Linear Relationships and Slope

Lesson: Slope from a Graph

When you look at a line on a coordinate plane, you can describe how steep it is and whether it goes up or down. That measurement is called the slope.

Slope tells us how much a line changes vertically compared to how much it changes horizontally. Another way to say this is rise over run.

In math, slope is written as:

$$\text{slope} = \frac{\text{rise}}{\text{run}}$$

If you can read points from a graph, you can find the slope of a line.

1. What slope means

• Rise means how far you move up or down.
• Run means how far you move left or right.
• Slope compares these two changes.

If a line goes up as you move to the right, the slope is positive.

If a line goes down as you move to the right, the slope is negative.

If a line is completely flat, the slope is 0.

If a line goes straight up and down, the slope is undefined because the run is 0.

2. How to find slope from a graph

To find slope from a graph, follow these steps:

1. Find two points on the line that land exactly on grid intersections.
2. Start at one point and move to the other point.
3. Count the rise: how many units up or down.
4. Count the run: how many units right.
5. Write the slope as $$\frac{\text{rise}}{\text{run}}$$ and simplify if needed.

It is usually easiest to move left to right. That helps you keep the sign correct.

3. Positive, negative, zero, and undefined slope

• Positive slope: line rises from left to right
• Negative slope: line falls from left to right
• Zero slope: horizontal line
• Undefined slope: vertical line

These are important because they tell you what the graph looks like even before you calculate the exact slope.

4. Using coordinates from the graph

If the graph shows two points, you can also use their coordinates. Suppose the points are \((x_1, y_1)\) and \((x_2, y_2)\). Then:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Here, \(m\) stands for slope. This is another way to say rise over run.

For 8th Grade, it is often easiest to count on the graph first, then check with the coordinates.

Worked Example 1: Positive slope

A line passes through the points \((1, 2)\) and \((4, 5)\).

Start at \((1, 2)\) and move to \((4, 5)\).

• Rise: from 2 to 5 is up 3
• Run: from 1 to 4 is right 3

So the slope is:

$$\frac{3}{3} = 1$$

Answer: The slope is \(1\).

This means the line goes up 1 unit for every 1 unit it moves right.

Worked Example 2: Negative slope

A line passes through the points \((2, 6)\) and \((6, 2)\).

Move from left to right, starting at \((2, 6)\).

• Rise: from 6 to 2 is down 4, so the rise is \(-4\)
• Run: from 2 to 6 is right 4, so the run is \(4\)

Now find the slope:

$$\frac{-4}{4} = -1$$

Answer: The slope is \(-1\).

This means the line goes down 1 unit for every 1 unit it moves right.

Worked Example 3: Fraction slope

A line passes through the points \((0, 1)\) and \((4, 3)\).

• Rise: from 1 to 3 is up 2
• Run: from 0 to 4 is right 4

So the slope is:

$$\frac{2}{4} = \frac{1}{2}$$

Answer: The slope is \(\frac{1}{2}\).

This means the line goes up 1 unit for every 2 units it moves right.

Worked Example 4: Zero and undefined slope

Horizontal line: Suppose a line passes through \((1, 4)\) and \((5, 4)\).

• Rise: from 4 to 4 is 0
• Run: from 1 to 5 is 4

$$\frac{0}{4} = 0$$

The slope is \(0\).

Vertical line: Suppose a line passes through \((3, 1)\) and \((3, 6)\).

• Rise: from 1 to 6 is 5
• Run: from 3 to 3 is 0

$$\frac{5}{0}$$

You cannot divide by 0, so the slope is undefined.

5. Important tips when reading a graph

• Choose points that are exactly on the line and on grid corners.
• Move in the same direction each time, usually left to right.
• If you move down, your rise is negative.
• If you use coordinates, subtract in the same order for top and bottom of the fraction.
• Simplify the fraction if possible.

6. Common mistakes to avoid

• Mixing up rise and run: rise is vertical, run is horizontal.
• Forgetting the sign: down means negative rise.
• Counting spaces incorrectly: count the units carefully on the grid.
• Using points not on the line: make sure both points are really on the line.
• Calling a vertical line slope 0: vertical lines have undefined slope, not 0.

7. Quick check: what kind of slope is it?

• Line goes up to the right \(\rightarrow\) positive
• Line goes down to the right \(\rightarrow\) negative
• Line is flat \(\rightarrow\) 0
• Line is straight up and down \(\rightarrow\) undefined

8. Practice thinking

If a line rises 5 and runs 2, the slope is:

$$\frac{5}{2}$$

If a line goes down 3 and right 6, the slope is:

$$\frac{-3}{6} = \frac{-1}{2}$$

If a line does not rise at all, the slope is:

$$0$$

Summary

Slope tells how steep a line is and whether it goes up or down. To find slope from a graph, pick two points on the line and count rise over run.

Remember:

• Positive slope: up to the right
• Negative slope: down to the right
• Zero slope: horizontal line
• Undefined slope: vertical line

If you carefully count vertical change and horizontal change, you can find the slope of any line shown on a graph.

The Slope Formula

When we look at a line on a coordinate plane, one important idea is its slope. Slope tells us how steep a line is and whether it goes up, down, or stays flat as we move from left to right.

The slope formula helps us find the slope when we know two points on a line.

The formula is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Here, m stands for slope. The points are written as \((x_1, y_1)\) and \((x_2, y_2)\).

This formula means:

• Subtract the y-values to find the change in y, also called the rise.
• Subtract the x-values to find the change in x, also called the run.
• Then divide.

So slope can also be thought of as:

$$\text{slope} = \frac{\text{rise}}{\text{run}}$$

Why order matters: If you use \(y_2 - y_1\) on top, then you must use \(x_2 - x_1\) on the bottom in the same order. If you switch the order for both, the slope will still be correct. But if you switch only the top or only the bottom, your answer will be wrong.

How to use the slope formula

1. Write the two points.
2. Label one point as \((x_1, y_1)\) and the other as \((x_2, y_2)\).
3. Substitute the numbers into the formula.
4. Simplify carefully.

Example 1: Positive slope

Find the slope of the line through \((1, 2)\) and \((5, 10)\).

Step 1: Use the formula.

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Let \((x_1, y_1) = (1, 2)\) and \((x_2, y_2) = (5, 10)\).

Step 2: Substitute.

$$m = \frac{10 - 2}{5 - 1}$$

Step 3: Simplify.

$$m = \frac{8}{4} = 2$$

The slope is 2.

This means the line goes up 2 units for every 1 unit it moves to the right.

Example 2: Negative slope

Find the slope of the line through \((3, 7)\) and \((8, 2)\).

Use \((x_1, y_1) = (3, 7)\) and \((x_2, y_2) = (8, 2)\).

$$m = \frac{2 - 7}{8 - 3}$$

$$m = \frac{-5}{5} = -1$$

The slope is -1.

A negative slope means the line goes down as you move from left to right.

Example 3: A fractional slope

Find the slope of the line through \((-2, 1)\) and \((4, 5)\).

Let \((x_1, y_1) = (-2, 1)\) and \((x_2, y_2) = (4, 5)\).

$$m = \frac{5 - 1}{4 - (-2)}$$

Notice that subtracting a negative becomes addition.

$$m = \frac{4}{6}$$

$$m = \frac{2}{3}$$

The slope is \(\frac{2}{3}\).

This means the line rises 2 units for every 3 units it moves to the right.

Example 4: Zero slope

Find the slope of the line through \((2, 4)\) and \((7, 4)\).

Use the formula:

$$m = \frac{4 - 4}{7 - 2}$$

$$m = \frac{0}{5} = 0$$

The slope is 0.

A slope of 0 means the line is horizontal. It does not go up or down.

Special case: Undefined slope

What if the x-values are the same? For example, the points \((3, 1)\) and \((3, 6)\).

$$m = \frac{6 - 1}{3 - 3} = \frac{5}{0}$$

Division by 0 is not possible, so the slope is undefined.

An undefined slope means the line is vertical.

What the sign of the slope tells you

• Positive slope: the line rises from left to right.
• Negative slope: the line falls from left to right.
• Zero slope: the line is horizontal.
• Undefined slope: the line is vertical.

Common mistakes to avoid

• Mixing the order: If you subtract the y-values in one order, subtract the x-values in that same order.
• Forgetting negative signs: Be careful when subtracting negative numbers.
• Switching x and y: The top is always the change in y, and the bottom is always the change in x.
• Dividing by zero: If the x-values are the same, the slope is undefined.

Quick check with reversed order

In Example 1, we used \((1, 2)\) and \((5, 10)\):

$$m = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$$

If we reverse the order of both points, we get:

$$m = \frac{2 - 10}{1 - 5} = \frac{-8}{-4} = 2$$

The slope is still 2. This shows that reversing both the top and bottom keeps the answer the same.

Summary

The slope formula is used to find the slope of a line from two points:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Slope tells how much a line rises or falls compared to how far it moves left or right. A positive slope rises, a negative slope falls, a slope of 0 is horizontal, and an undefined slope is vertical.

When using the formula, subtract carefully and keep the order of the points consistent. With practice, the slope formula becomes a quick and useful tool for understanding linear relationships.

Similar Triangles and Constant Slope

When we draw a straight, non-vertical line on a coordinate plane, the line has a special property: its slope stays the same no matter which two points on the line we choose.

This lesson explains why that happens. We will use similar triangles to show that the ratio of vertical change to horizontal change is constant along a line.

Remember:

• Rise = how much the graph goes up or down
• Run = how much the graph goes left or right
• Slope = $$\frac{\text{rise}}{\text{run}}$$

For a non-vertical line, the run is never 0, so the slope is defined.

1. Review: What are similar triangles?

Two triangles are similar if they have the same shape, even if they are different sizes.

In similar triangles:

• Matching angles are equal.
• Matching side lengths have the same ratio.

For example, if one right triangle has legs 3 and 4, and a larger similar triangle has legs 6 and 8, then the side lengths were multiplied by 2. The triangles are different sizes, but the ratio of leg lengths stays the same:

$$\frac{3}{4}=\frac{6}{8}$$

This idea of equal ratios is the key to understanding slope.

2. Making triangles on a line

Suppose a straight line passes through several points. If we pick any two points on the line, we can draw a right triangle by moving horizontally from one point and vertically to the other point.

The horizontal side shows the run. The vertical side shows the rise.

If we choose a different pair of points on the same line, we can make another right triangle.

These right triangles will have:

• a right angle, because horizontal and vertical segments meet at 90 degrees,
• and the same acute angle where the line meets the horizontal side.

Because they have two matching angles, the triangles are similar.

That means the ratio

$$\frac{\text{rise}}{\text{run}}$$

is the same for both triangles.

So the slope is constant all along the line.

3. Why slope is constant on a non-vertical line

Let two different triangles be formed on the same line.

Triangle 1 has rise \(r_1\) and run \(u_1\).

Triangle 2 has rise \(r_2\) and run \(u_2\).

Since the triangles are similar, corresponding side lengths are proportional:

$$\frac{r_1}{u_1}=\frac{r_2}{u_2}$$

But each of these ratios is just slope. So both triangles give the same slope.

This proves an important fact:

For any two points on the same non-vertical line, the slope is always the same.

4. Connecting this to the slope formula

If a line passes through points \((x_1,y_1)\) and \((x_2,y_2)\), then:

$$\text{rise}=y_2-y_1$$

$$\text{run}=x_2-x_1$$

So the slope formula is:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

This formula works because of the similar triangles idea. No matter which two points you choose on the line, the ratio stays the same.

5. Worked Example 1: Finding slope from one triangle

A line goes through the points \((1,2)\) and \((5,6)\). Find the slope.

Step 1: Find the rise.

$$6-2=4$$

Step 2: Find the run.

$$5-1=4$$

Step 3: Write rise over run.

$$m=\frac{4}{4}=1$$

Answer: The slope is \(1\).

This means the line goes up 1 unit for every 1 unit it moves right.

6. Worked Example 2: Showing slope stays the same with different points

Suppose the same line contains the points \((1,2)\), \((3,4)\), and \((5,6)\).

Let us find the slope using two different pairs of points.

Using \((1,2)\) and \((3,4)\):

$$m=\frac{4-2}{3-1}=\frac{2}{2}=1$$

Using \((1,2)\) and \((5,6)\):

$$m=\frac{6-2}{5-1}=\frac{4}{4}=1$$

Both slopes are the same.

Why? The triangles made from these points are similar. One triangle may be larger, but the ratio of rise to run does not change.

7. Worked Example 3: A line with negative slope

A line passes through \((2,7)\) and \((6,3)\). Find the slope.

Step 1: Find the rise.

$$3-7=-4$$

Step 2: Find the run.

$$6-2=4$$

Step 3: Divide.

$$m=\frac{-4}{4}=-1$$

Answer: The slope is \(-1\).

A negative slope means the line goes downward as you move from left to right.

If you made larger or smaller triangles on this same line, they would still be similar, and each one would still give slope \(-1\).

8. Worked Example 4: Using similar triangles to compare slopes

On one line, a small right triangle has rise 3 and run 5. A larger triangle on the same line has run 15. What is its rise?

Since both triangles are on the same line, they are similar. So their rise-to-run ratios are equal:

$$\frac{3}{5}=\frac{x}{15}$$

Now solve for \(x\).

Because \(15\) is 3 times \(5\), the rise must also be 3 times \(3\):

$$x=9$$

Answer: The larger triangle has rise 9.

Check the slope:

$$\frac{3}{5}=\frac{9}{15}$$

The ratios match, so the slope stays constant.

9. Important idea: why we say non-vertical line

A vertical line has no horizontal change, so its run is 0.

That would make the slope:

$$\frac{\text{rise}}{0}$$

Division by 0 is not allowed, so vertical lines have undefined slope.

That is why this idea of constant slope is stated for non-vertical lines.

10. Common mistakes to avoid

• Mixing up rise and run. Rise is the change in \(y\). Run is the change in \(x\).
• Subtracting in different orders. If you do \(y_2-y_1\), then you must also do \(x_2-x_1\).
• Forgetting negative slope. If the line goes down as you move right, the slope is negative.
• Using a vertical line. If the run is 0, the slope is undefined.

11. What this means for linear relationships

A linear relationship makes a straight line when graphed.

The constant slope tells us the line changes at a steady rate. For example, if the slope is 2, then every time \(x\) increases by 1, \(y\) increases by 2.

This steady change is what makes the graph a straight line instead of a curve.

12. Quick check for understanding

1. If two right triangles are drawn on the same non-vertical line, why are they similar?
2. If one triangle on a line has rise 4 and run 7, what is the slope?
3. If another triangle on that same line has run 14, what must its rise be?
4. What kind of line has undefined slope?

Answers:

1. They both have a right angle and share the same angle made by the line, so they have two equal angles.
2. $$\frac{4}{7}$$
3. Since the run doubled from 7 to 14, the rise also doubles from 4 to 8.
4. A vertical line.

Summary

On any straight, non-vertical line, triangles formed by rise and run are similar triangles.

Because similar triangles have equal side ratios, the ratio

$$\frac{\text{rise}}{\text{run}}$$

is always the same.

That ratio is the slope, so the slope of a non-vertical line is constant between any two points on the line.

Lesson: The Y-Intercept

When we graph a line, one important number to look for is the y-intercept. The y-intercept tells us where the line crosses the y-axis.

This is useful because it gives us a starting point for the line. In real-world situations, the y-intercept often represents the starting amount or initial value before anything changes.

In this lesson, you will learn:

• what the y-intercept means on a graph,
• how to find it in an equation,
• how to write it as a point, and
• how to interpret it in real-life situations.

1. What is the y-intercept?

The y-intercept is the point where a line crosses the y-axis.

Every point on the y-axis has an x-coordinate of 0. So to find the y-intercept, we look for the point on the line where:

$$x = 0$$

If a line crosses the y-axis at 5, then its y-intercept is the point \,\((0,5)\).

If a line crosses the y-axis at \(-2\), then its y-intercept is \,\((0,-2)\).

2. Finding the y-intercept from an equation

A very common way to write a linear equation is:

$$y = mx + b$$

This is called slope-intercept form.

• \(m\) is the slope, which tells how steep the line is.
• \(b\) is the y-intercept, which tells where the line crosses the y-axis.

So in the equation \(y = mx + b\), the y-intercept is just the number \(b\).

For example:

• In \(y = 2x + 3\), the y-intercept is \(3\), so the point is \((0,3)\).
• In \(y = -4x + 1\), the y-intercept is \(1\), so the point is \((0,1)\).
• In \(y = \frac{1}{2}x - 6\), the y-intercept is \(-6\), so the point is \((0,-6)\).

3. Why is the x-coordinate always 0?

The y-axis is the vertical axis on a coordinate plane. Any point on that axis is not to the left or right of the origin, so its x-value must be 0.

That means every y-intercept has this form:

$$ (0,b) $$

Here, \(b\) is the y-value where the line crosses the y-axis.

4. Finding the y-intercept by substituting \(x=0\)

Even if an equation is not already written as \(y = mx + b\), you can still find the y-intercept by plugging in \(x=0\).

This works because the y-intercept is the point where the graph crosses the y-axis, and on the y-axis, \(x=0\).

For example, if the equation is:

$$y = 3x - 7$$

Substitute \(x=0\):

$$y = 3(0) - 7 = -7$$

So the y-intercept is:

$$ (0,-7) $$

5. The y-intercept on a graph

On a graph, the y-intercept is the point where the line touches or crosses the y-axis.

To find it:

1. Locate the y-axis.
2. Find where the line crosses it.
3. Write that point as \((0,y)\).

For example, if the line crosses the y-axis at \(4\), the y-intercept is \((0,4)\).

If it crosses at \(-3\), the y-intercept is \((0,-3)\).

6. The y-intercept as an initial value

In many word problems, the y-intercept represents the amount you start with before the value begins to increase or decrease.

For example, suppose a taxi ride costs a starting fee of \(\$5\), and then \(\$2\) for each mile. The total cost \(y\) after \(x\) miles can be written as:

$$y = 2x + 5$$

Here, the y-intercept is \(5\). That means when \(x=0\) miles, the cost is \(\$5\). This is the starting cost.

This is why the y-intercept is often called the initial value.

Worked Example 1: Find the y-intercept from slope-intercept form

Find the y-intercept of:

$$y = 4x + 2$$

Step 1: Compare the equation to \(y = mx + b\).

Here, \(b=2\).

Step 2: Write the intercept as a point.

Since y-intercepts are on the y-axis, \(x=0\).

So the y-intercept is:

$$ (0,2) $$

Answer: The y-intercept is \(2\), or as a point, \((0,2)\).

Worked Example 2: Find the y-intercept with a negative value

Find the y-intercept of:

$$y = -3x - 5$$

Step 1: Identify \(b\).

The equation is already in the form \(y = mx + b\), so \(b=-5\).

Step 2: Write the point.

$$ (0,-5) $$

Answer: The y-intercept is \(-5\), and the point is \((0,-5)\).

Worked Example 3: Find the y-intercept by substituting \(x=0\)

Find the y-intercept of:

$$y = 6x + 1$$

You could read it directly, but let us use substitution to practice.

Step 1: Let \(x=0\).

$$y = 6(0) + 1$$

Step 2: Simplify.

$$y = 1$$

Step 3: Write the point.

$$ (0,1) $$

Answer: The y-intercept is \((0,1)\).

Worked Example 4: Interpret the y-intercept in a real-world situation

A gym charges \(\$20\) to sign up and then \(\$15\) each month. The total cost after \(x\) months is:

$$y = 15x + 20$$

What is the y-intercept, and what does it mean?

Step 1: Identify \(b\).

In \(y = 15x + 20\), the y-intercept is \(20\).

Step 2: Write it as a point.

$$ (0,20) $$

Step 3: Interpret it.

When \(x=0\) months, the total cost is \(\$20\). This means the starting cost is \(\$20\), which is the sign-up fee.

Answer: The y-intercept is \((0,20)\), and it represents the starting fee.

Common mistakes to avoid

• Mixing up the slope and the y-intercept. In \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept.
• Forgetting that the y-intercept is a point on the y-axis. The x-coordinate must always be 0.
• Writing the intercept as \((b,0)\). That would be an x-intercept, not a y-intercept.
• Ignoring negative signs. In \(y = 2x - 4\), the y-intercept is \(-4\), not \(4\).

Quick check

Try these on your own:

• For \(y = x + 7\), what is the y-intercept?
• For \(y = -2x + 9\), where does the line cross the y-axis?
• In \(y = 5x - 3\), what is the initial value?

Answers:

• \((0,7)\)
• \((0,9)\)
• \(-3\)

Summary

The y-intercept is where a line crosses the y-axis. Since every point on the y-axis has \(x=0\), the y-intercept is always found when \(x=0\).

In the equation \(y = mx + b\), the number \(b\) is the y-intercept. Written as a point, it is \((0,b)\).

In real-world problems, the y-intercept often tells the starting value or initial amount. Understanding the y-intercept helps you read graphs, understand equations, and make sense of situations involving linear relationships.

Slope-Intercept Form is one of the most useful ways to write a linear equation. It helps us understand a line quickly by showing its slope and its y-intercept.

The slope-intercept form of a line is:

$$y = mx + b$$

In this form:

• (y) is the output value.
• (x) is the input value.
• (m) is the slope, or rate of change.
• (b) is the y-intercept, or where the line crosses the y-axis.

When a line is written in slope-intercept form, you can learn two important things right away:

• How steep the line is and whether it goes up or down.
• Where the line starts on the y-axis.

Lets look more closely at each part.

1. The slope (m)

Slope tells how much the line changes as you move from left to right. Another way to say this is that slope is the rise over run.

$$m = \frac{\text{change in } y}{\text{change in } x}$$

If the slope is positive, the line rises from left to right. If the slope is negative, the line falls from left to right.

• If \(m = 2\), the line goes up 2 for every 1 right.
• If \(m = -3\), the line goes down 3 for every 1 right.
• If \(m = \frac{1}{2}\), the line goes up 1 for every 2 right.

2. The y-intercept (b)

The y-intercept is the point where the line crosses the y-axis. On the y-axis, the x-value is always 0, so the y-intercept is always a point of the form \((0, b)\).

For example:

• If \(b = 4\), the line crosses the y-axis at \((0, 4)\).
• If \(b = -2\), the line crosses the y-axis at \((0, -2)\).

How to read an equation in slope-intercept form

Suppose you see the equation:

$$y = 3x + 1$$

This tells you:

• The slope is \(3\).
• The y-intercept is \(1\), so the line crosses the y-axis at \((0,1)\).

To graph this line:

1. Plot the y-intercept \((0,1)\).
2. Use the slope \(3\), which means \(\frac{3}{1}\).
3. From \((0,1)\), move up 3 and right 1 to get another point.
4. Draw a straight line through the points.

Worked Example 1: Identify slope and y-intercept

Find the slope and y-intercept of:

$$y = 5x - 2$$

Step 1: Compare the equation to \(y = mx + b\).

Here, \(m = 5\) and \(b = -2\).

Answer:

• Slope: \(5\)
• y-intercept: \(-2\), or the point \((0,-2)\)

Worked Example 2: Graph from slope-intercept form

Graph the line:

$$y = -2x + 3$$

Step 1: Identify the slope and y-intercept.

• \(m = -2\)
• \(b = 3\)

Step 2: Plot the y-intercept at \((0,3)\).

Step 3: Use the slope \(-2\). Write it as:

$$-2 = \frac{-2}{1}$$

This means go down 2 and right 1.

Starting at \((0,3)\):

• Go down 2 to \(1\)
• Go right 1 to \(1\)

So another point is \((1,1)\).

You could do it again to get another point, such as \((2,-1)\).

Answer: Plot \((0,3)\), \((1,1)\), and \((2,-1)\), then draw the line.

Worked Example 3: Write an equation from slope and y-intercept

Write the equation of a line with slope \(\frac{1}{2}\) and y-intercept \(-4\).

Step 1: Start with the form:

$$y = mx + b$$

Step 2: Substitute the given values.

• \(m = \frac{1}{2}\)
• \(b = -4\)

$$y = \frac{1}{2}x - 4$$

Answer: The equation is \(y = \frac{1}{2}x - 4\).

Worked Example 4: Write an equation from a graph description

A line crosses the y-axis at \((0,2)\). From that point, it goes up 1 and right 3. Write the equation.

Step 1: Find the y-intercept.

The line crosses at \((0,2)\), so \(b = 2\).

Step 2: Find the slope.

It goes up 1 and right 3, so:

$$m = \frac{1}{3}$$

Step 3: Use slope-intercept form.

$$y = mx + b$$

$$y = \frac{1}{3}x + 2$$

Answer: The equation is \(y = \frac{1}{3}x + 2\).

How slope-intercept form connects to real life

Slope-intercept form can describe situations where one amount changes at a constant rate and starts with an initial amount.

For example, suppose a bike rental costs \(\$4\) to start and then \(\$2\) for each hour.

Let:

• \(x\) = number of hours
• \(y\) = total cost

The equation is:

$$y = 2x + 4$$

In this equation:

• The slope \(2\) means the cost increases by \(\$2\) each hour.
• The y-intercept \(4\) means the starting cost is \(\$4\).

This is why slope is often called the rate of change, and the y-intercept is often called the initial value.

Common mistakes to avoid

• Mixing up slope and y-intercept. In \(y = mx + b\), the number with \(x\) is the slope. The number by itself is the y-intercept.
• Forgetting the sign. In \(y = -3x + 5\), the slope is \(-3\), not \(3\).
• Thinking \(b\) is an x-value. The y-intercept is a y-value, and the point is \((0,b)\).
• Not writing the slope as a fraction when graphing. A whole number like \(4\) can be written as \(\frac{4}{1}\).

Quick review

• Slope-intercept form is $$y = mx + b$$
• \(m\) is the slope, or rate of change
• \(b\) is the y-intercept, or initial value
• The y-intercept is the point \((0,b)\)
• You can graph a line by plotting the y-intercept and then using the slope

Summary

Slope-intercept form makes linear equations easier to read, write, and graph. When you see \(y = mx + b\), you can immediately identify the slope and the y-intercept. This helps you understand how the line changes and where it begins on the coordinate plane.

Point-Slope Form is a way to write the equation of a line when you know the slope and one point on the line.

This is helpful because sometimes you are not given the y-intercept. Instead, you may be told a slope and a point like \,\((3, 5)\). Point-slope form lets you write the equation directly from that information.

The point-slope form of a line is:

$$y - y_1 = m(x - x_1)$$

In this formula:

• \(m\) is the slope
• \((x_1, y_1)\) is a point on the line

So if you know the slope and one point, you can substitute those values into the formula.

Why does this form make sense?

Remember that slope tells us how steep a line is. Slope compares the change in y-values to the change in x-values.

If you start at a known point \,\((x_1, y_1)\) and move to another point \,\((x, y)\) on the same line, then the slope between those two points must still be \,\(m\).

That idea leads to point-slope form. You do not need to memorize where it comes from right now, but it helps to know that it is based on the meaning of slope.

How to use point-slope form

1. Identify the slope \,\(m\).
2. Identify the point \,\((x_1, y_1)\).
3. Substitute into \,\(y - y_1 = m(x - x_1)\).
4. If needed, simplify to slope-intercept form \,\(y = mx + b\).

Important sign reminder: When substituting a point, pay close attention to negatives.

For example, if the point is \,\((-2, 4)\), then:

$$y - 4 = m(x - (-2))$$

This becomes:

$$y - 4 = m(x + 2)$$

Subtracting a negative turns into adding.

Worked Example 1: Write an equation from a slope and a point

Write the equation of the line with slope \,\(m = 3\) that passes through \,\((2, 1)\).

Step 1: Use point-slope form

$$y - y_1 = m(x - x_1)$$

Step 2: Substitute

Here, \,\(m = 3\), \,\(x_1 = 2\), and \,\(y_1 = 1\).

$$y - 1 = 3(x - 2)$$

That equation is already correct in point-slope form.

Step 3: Change to slope-intercept form

Distribute the 3:

$$y - 1 = 3x - 6$$

Add 1 to both sides:

$$y = 3x - 5$$

So the equation can be written as:

• Point-slope form: \,\(y - 1 = 3(x - 2)\)
• Slope-intercept form: \,\(y = 3x - 5\)

Worked Example 2: Using a negative y-value

Write the equation of the line with slope \,\(m = -2\) passing through \,\((4, -3)\).

Step 1: Substitute into point-slope form

$$y - y_1 = m(x - x_1)$$

$$y - (-3) = -2(x - 4)$$

Simplify the left side:

$$y + 3 = -2(x - 4)$$

Step 2: Change to slope-intercept form

Distribute \,\(-2\):

$$y + 3 = -2x + 8$$

Subtract 3 from both sides:

$$y = -2x + 5$$

So the equation is:

• Point-slope form: \,\(y + 3 = -2(x - 4)\)
• Slope-intercept form: \,\(y = -2x + 5\)

Worked Example 3: Using a negative x-value

Write the equation of the line with slope \,\(m = \frac{1}{2}\) passing through \,\((-6, 7)\).

Step 1: Substitute carefully

$$y - 7 = \frac{1}{2}(x - (-6))$$

Because \,\(x_1 = -6\), the expression \,\(x - (-6)\) becomes \,\(x + 6\).

$$y - 7 = \frac{1}{2}(x + 6)$$

Step 2: Change to slope-intercept form

Distribute \,\(\frac{1}{2}\):

$$y - 7 = \frac{1}{2}x + 3$$

Add 7 to both sides:

$$y = \frac{1}{2}x + 10$$

So the equation is:

• Point-slope form: \,\(y - 7 = \frac{1}{2}(x + 6)\)
• Slope-intercept form: \,\(y = \frac{1}{2}x + 10\)

Worked Example 4: Start from words

A line has slope \,\(4\) and passes through the point \,\((0, -2)\). Write the equation.

Step 1: Identify what you know

• Slope: \,\(m = 4\)
• Point: \,\((0, -2)\)

Step 2: Substitute into point-slope form

$$y - (-2) = 4(x - 0)$$

Simplify:

$$y + 2 = 4x$$

Step 3: Solve for y

$$y = 4x - 2$$

This example shows that even if the point has \,\(x = 0\), point-slope form still works.

How point-slope form connects to slope-intercept form

Point-slope form and slope-intercept form both describe the same line. They are just different ways of writing the equation.

• Point-slope form: \,\(y - y_1 = m(x - x_1)\)
• Slope-intercept form: \,\(y = mx + b\)

Point-slope form is best when you know:

• the slope
• one point on the line

Slope-intercept form is best when you want to easily see:

• the slope \,\(m\)
• the y-intercept \,\(b\)

Common mistakes to avoid

• Mixing up the point values: In \,\((x_1, y_1)\), the first number is always x and the second number is always y.
• Forgetting parentheses: Always write \,\((x - x_1)\) in parentheses before simplifying.
• Sign mistakes with negatives: For example, if \,\(x_1 = -3\), then \,\(x - (-3) = x + 3\).
• Changing the slope by mistake: The slope \,\(m\) stays exactly the same when you substitute.

Quick check

If the slope is \,\(5\) and the point is \,\((1, 8)\), then the point-slope form is:

$$y - 8 = 5(x - 1)$$

If the slope is \,\(-1\) and the point is \,\((-2, 3)\), then the point-slope form is:

$$y - 3 = -1(x - (-2))$$

which simplifies to

$$y - 3 = -1(x + 2)$$

Summary

Point-slope form is a useful way to write the equation of a line when you know the slope and one point.

The formula is:

$$y - y_1 = m(x - x_1)$$

To use it, substitute the slope and the point carefully, especially when negative numbers are involved.

Then, if needed, simplify the equation to slope-intercept form \,\(y = mx + b\).

Standard Form of a Linear Equation is one way to write the equation of a line. In 8th Grade, the standard form is usually written as:

In this form, A, B, and C are numbers, and A and B are not both 0.

This form is useful because it makes it easier to find the x-intercept and y-intercept of a line. Intercepts are the points where the line crosses the axes.

• The x-intercept is where the line crosses the x-axis. At that point, the y-value is 0.
• The y-intercept is where the line crosses the y-axis. At that point, the x-value is 0.

Learning standard form helps you move between different equation forms and understand how a line behaves on the coordinate plane.

Main Idea: What standard form looks like

An equation is in standard form if the x-term and y-term are on the same side, and the number is on the other side.

For example:

• \(3x + 2y = 12\) is in standard form.
• \(5x - y = 10\) is in standard form.
• \(y = 2x + 4\) is not in standard form. It is in slope-intercept form.

Sometimes the equation may need to be rearranged so it matches the pattern \(Ax + By = C\).

How to write an equation in standard form

1. Put the x-term and y-term on the same side of the equation.
2. Put the constant number on the other side.
3. If needed, combine like terms.
4. Try to keep the coefficients as whole numbers.

For example, start with:

Move the x-term to the left side by subtracting \(2x\) from both sides:

This matches standard form. Another common version is to make the x-term positive:

Both equations represent the same line.

Finding intercepts from standard form

One big reason standard form is helpful is that intercepts are easy to find.

• To find the x-intercept, let \(y = 0\).
• To find the y-intercept, let \(x = 0\).

Then solve for the remaining variable.

Worked Example 1: Find the intercepts of a line in standard form

Find the x-intercept and y-intercept of:

Step 1: Find the x-intercept

Set \(y = 0\):

So the x-intercept is \((6, 0)\).

Step 2: Find the y-intercept

Set \(x = 0\):

So the y-intercept is \((0, 4)\).

Worked Example 2: Rewrite an equation in standard form

Write this equation in standard form:

We want the x-term and y-term on the same side. Subtract \(4x\) from both sides:

This is standard form.

If you want the x-term to be positive, multiply every term by \(-1\):

So a standard form of the equation is:

Worked Example 3: Find intercepts after rewriting the equation

Find the intercepts of:

Step 1: Rewrite in standard form

Add \(3x\) to both sides:

Step 2: Find the x-intercept

Set \(y = 0\):

The x-intercept is \((3, 0)\).

Step 3: Find the y-intercept

Set \(x = 0\):

The y-intercept is \((0, 9)\).

Worked Example 4: Standard form with subtraction

Find the intercepts of:

x-intercept: Let \(y = 0\).

The x-intercept is \((4, 0)\).

y-intercept: Let \(x = 0\).

The y-intercept is \((0, -10)\).

Tips to remember

• Standard form looks like \(Ax + By = C\).
• To find the x-intercept, always substitute \(y = 0\).
• To find the y-intercept, always substitute \(x = 0\).
• If the equation is not in standard form, rearrange it first.
• Check that your intercept points make sense: x-intercepts have y-value 0, and y-intercepts have x-value 0.

Common mistakes

• Mixing up the intercept rules: For the x-intercept, use \(y = 0\), not \(x = 0\).
• Forgetting to move terms correctly: When rewriting an equation, do the same operation to both sides.
• Not writing intercepts as points: Write \((6, 0)\), not just \(6\).
• Sign mistakes: Be careful with negative numbers when solving.

Quick Check

Try these on your own:

1. Write \(y = 2x + 5\) in standard form.
2. Find the x-intercept of \(4x + y = 8\).
3. Find the y-intercept of \(6x - 3y = 12\).

Answers:

1. \(2x - y = -5\) or \(-2x + y = 5\)
2. Set \(y = 0\): \(4x = 8\), so \(x = 2\). The x-intercept is \((2, 0)\).
3. Set \(x = 0\): \(-3y = 12\), so \(y = -4\). The y-intercept is \((0, -4)\).

Summary

Standard form is a way to write a linear equation as \(Ax + By = C\). This form is especially helpful for finding intercepts quickly. To find the x-intercept, set \(y = 0\). To find the y-intercept, set \(x = 0\). With practice, you will be able to rewrite equations and find intercepts with confidence.

Graphing from Various Forms

Linear equations can be written in different forms, but they can all describe the same kind of graph: a straight line.

In this lesson, you will learn how to graph a line when the equation is written in different ways. Sometimes the equation shows the slope and y-intercept. Sometimes it is in standard form, and sometimes it gives you a point and a slope.

The big idea is this: no matter what form the equation is in, you need enough information to plot points and draw the line.

Important vocabulary

• Slope: how steep a line is; it tells how much the line rises or falls compared to how much it moves right.
• y-intercept: the point where the line crosses the y-axis.
• x-intercept: the point where the line crosses the x-axis.
• Graph: the picture of the equation on the coordinate plane.

1. Graphing from slope-intercept form

Slope-intercept form looks like this:

Here:

• \(m\) is the slope.
• \(b\) is the y-intercept.

This form is usually the easiest to graph because it tells you where the line starts and how it moves.

How to graph from slope-intercept form

1. Find the y-intercept \((0,b)\).
2. Plot that point on the y-axis.
3. Use the slope \(m\) to find another point.
4. Draw a straight line through the points.

Remember that slope is often written as a fraction:

For example, if the slope is \(\frac{2}{3}\), go up 2 and right 3.

If the slope is negative, one direction will be down. For example, \(-\frac{1}{2}\) means go down 1 and right 2.

Worked Example 1: Slope-intercept form

Graph the line:

Step 1: Find the y-intercept.

The y-intercept is \(1\), so plot the point \((0,1)\).

Step 2: Use the slope.

The slope is \(\frac{2}{3}\). From \((0,1)\), go up 2 and right 3.

That gives another point: \((3,3)\).

You could also repeat the slope again to get another point, such as \((6,5)\).

Step 3: Draw the line.

Connect the points with a straight line.

2. Graphing from standard form

Standard form looks like this:

For 8th grade, a very useful way to graph from standard form is to find the intercepts.

How to find intercepts

• To find the x-intercept, let \(y=0\).
• To find the y-intercept, let \(x=0\).

Why does this work?

• Any point on the x-axis has a y-value of \(0\).
• Any point on the y-axis has an x-value of \(0\).

How to graph from standard form using intercepts

1. Set \(y=0\) and solve for \(x\). This gives the x-intercept.
2. Set \(x=0\) and solve for \(y\). This gives the y-intercept.
3. Plot both intercepts.
4. Draw a straight line through them.

Worked Example 2: Standard form

Graph the line:

Step 1: Find the x-intercept.

Let \(y=0\):

So the x-intercept is \((3,0)\).

Step 2: Find the y-intercept.

Let \(x=0\):

So the y-intercept is \((0,2)\).

Step 3: Plot the intercepts.

Plot \((3,0)\) and \((0,2)\).

Step 4: Draw the line.

Connect the points with a straight line.

Tip: If you want, you can also rewrite standard form into slope-intercept form first. But using intercepts is often faster.

3. Graphing from point-slope form

Point-slope form looks like this:

Here:

• \(m\) is the slope.
• \((x_1,y_1)\) is a point on the line.

This form tells you one point on the line and how the line moves from that point.

How to graph from point-slope form

1. Identify the given point \((x_1,y_1)\).
2. Plot that point.
3. Use the slope to find another point.
4. Draw the line through the points.

Worked Example 3: Point-slope form

Graph the line:

Step 1: Identify the point.

The point is \((1,2)\).

Step 2: Plot the point.

Place a point at \((1,2)\).

Step 3: Use the slope.

The slope is \(-\frac{1}{2}\).

This means go down 1 and right 2.

Starting at \((1,2)\):

• right 2 gives \(x=3\)
• down 1 gives \(y=1\)

So another point is \((3,1)\).

You could also move the opposite way: up 1 and left 2, giving the point \((-1,3)\).

Step 4: Draw the line.

Connect the points with a straight line.

4. Choosing a graphing strategy

Different forms suggest different graphing methods.

• If the equation is in \(y=mx+b\), use the y-intercept and slope.
• If the equation is in \(Ax+By=C\), use the x- and y-intercepts.
• If the equation is in \(y-y_1=m(x-x_1)\), use the given point and slope.

No matter which form you start with, graphing always comes down to plotting points correctly.

Worked Example 4: Comparing forms

Graph the line:

This is in slope-intercept form.

Step 1: Find the y-intercept.

The y-intercept is \(4\), so plot \((0,4)\).

Step 2: Use the slope.

The slope is \(-2\), which can be written as \(-\frac{2}{1}\).

From \((0,4)\), go down 2 and right 1 to get \((1,2)\).

Repeat to get another point: \((2,0)\).

Step 3: Draw the line.

Draw a straight line through \((0,4)\), \((1,2)\), and \((2,0)\).

Notice something important: the point \((2,0)\) is the x-intercept. Even when you graph using slope and y-intercept, you may notice other useful points along the way.

Common mistakes to avoid

• Mixing up rise and run. Rise is the vertical change; run is the horizontal change.
• Forgetting the sign of the slope. A negative slope means the line goes down as you move right.
• Using the wrong intercept. The y-intercept must be on the y-axis, and the x-intercept must be on the x-axis.
• Reading the point wrong in point-slope form. In \(y-y_1=m(x-x_1)\), the point is \((x_1,y_1)\).
• Drawing a curved line. Linear equations always graph as straight lines.

Quick check questions

1. In \(y=3x-2\), what is the y-intercept?
2. In \(y=3x-2\), what is the slope?
3. For \(x+2y=8\), what do you set equal to \(0\) to find the x-intercept?
4. In \(y-5=\frac{1}{4}(x+2)\), what point is given?

Answers

1. The y-intercept is \(-2\), so the point is \((0,-2)\).
2. The slope is \(3\).
3. Set \(y=0\).
4. The point is \((-2,5)\).

Summary

Linear equations can be graphed from different forms.

• In slope-intercept form, graph using the y-intercept and slope.
• In standard form, graph by finding the x-intercept and y-intercept.
• In point-slope form, graph using the given point and the slope.

If you can identify what information the equation gives you, you can graph the line correctly.

Horizontal and Vertical Lines are special kinds of lines on the coordinate plane. They look simple, but they are very important when learning about slope and graphing equations.

In this lesson, you will learn how to recognize horizontal and vertical lines, how their equations are written, and how their slopes are different.

Remember that on a coordinate plane, the x-axis goes left and right, and the y-axis goes up and down. A point is written as \((x, y)\).

A horizontal line goes straight across, from left to right. Every point on a horizontal line has the same y-value.

For example, on the line \(y = 3\), some points are \(( -2, 3)\), \((0, 3)\), and \((5, 3)\). The x-values change, but the y-value stays 3.

That means the equation of any horizontal line has the form:

$$y = b$$

where \(b\) is a number. It tells you the height of the line on the graph.

A vertical line goes straight up and down. Every point on a vertical line has the same x-value.

For example, on the line \(x = -4\), some points are \((-4, 2)\), \((-4, 0)\), and \((-4, -5)\). The y-values change, but the x-value stays \(-4\).

That means the equation of any vertical line has the form:

$$x = a$$

where \(a\) is a number. It tells you where the line is placed left or right on the graph.

Now let’s connect this to slope. Slope tells how steep a line is. A common slope formula is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

For a horizontal line, the y-values are the same. So the change in y is 0. That makes the slope:

$$m = \frac{0}{x_2 - x_1} = 0$$

So, horizontal lines have slope 0.

For a vertical line, the x-values are the same. So the change in x is 0. That would make the slope look like dividing by 0:

$$m = \frac{y_2 - y_1}{0}$$

But division by 0 is not allowed. So, vertical lines have undefined slope.

This is a very important difference:

• Horizontal line  slope is 0
• Vertical line  slope is undefined

It is also helpful to compare these lines to the axes:

• The x-axis is a horizontal line with equation \(y = 0\).
• The y-axis is a vertical line with equation \(x = 0\).

When graphing a horizontal line like \(y = 2\), start at 2 on the y-axis. Then draw a line straight across the graph.

When graphing a vertical line like \(x = -3\), start at \(-3\) on the x-axis. Then draw a line straight up and down.

One common mistake is mixing up \(x = a\) and \(y = b\). A good memory trick is:

• \(y = b\): stay at the same height  horizontal line
• \(x = a\): stay at the same side-to-side position  vertical line

Worked Example 1: Identify the type of line

What kind of line is \(y = -1\)? What is its slope?

Step 1: Notice that the equation is written as \(y = b\).

Step 2: Equations of the form \(y = b\) are horizontal lines.

Step 3: Horizontal lines have slope 0.

Answer: \(y = -1\) is a horizontal line, and its slope is 0.

Worked Example 2: Identify the type of line

What kind of line is \(x = 5\)? What is its slope?

Step 1: Notice that the equation is written as \(x = a\).

Step 2: Equations of the form \(x = a\) are vertical lines.

Step 3: Vertical lines have undefined slope.

Answer: \(x = 5\) is a vertical line, and its slope is undefined.

Worked Example 3: Write the equation of a line from a graph description

A line passes through the point \((3, 4)\) and is horizontal. What is its equation?

Step 1: A horizontal line keeps the same y-value.

Step 2: The point \((3, 4)\) has y-value 4.

Step 3: So the equation is:

$$y = 4$$

Answer: The equation is \(y = 4\).

Worked Example 4: Write the equation of a line from points

Find the equation of the line through \((-2, 1)\) and \((-2, 6)\).

Step 1: Compare the x-values and y-values.

The x-values are both \(-2\). The y-values are different.

Step 2: If the x-value stays the same, the line is vertical.

Step 3: A vertical line has equation \(x = a\), where \(a\) is the constant x-value.

So the equation is:

$$x = -2$$

Answer: The equation is \(x = -2\).

How to tell horizontal and vertical lines apart

1. Look at the equation.
2. If it is \(y = \text{number}\), the line is horizontal.
3. If it is \(x = \text{number}\), the line is vertical.
4. Then decide the slope:
   • horizontal  slope 0
   • vertical  slope undefined

Quick Check

• \(y = 7\)  horizontal or vertical? Horizontal
• \(x = -1\)  horizontal or vertical? Vertical
• Slope of \(y = 0\)? 0
• Slope of \(x = 0\)? Undefined

Summary

Horizontal lines go left and right and have equations of the form \(y = b\). Their slope is 0 because the y-value does not change.

Vertical lines go up and down and have equations of the form \(x = a\). Their slope is undefined because calculating slope would require dividing by 0.

If you remember same y means horizontal and same x means vertical, you will be able to graph and identify these lines with confidence.