Writing Equations in Slope-Intercept Form

To write an equation in slope-intercept form, find the slope m and the y-intercept b, then put them into $y = mx + b$ . If you don’t know b, substitute the slope and one point into $y = mx + b$ and solve for b. This works for a graph, two points or a word problem.

Before this lesson: You should know slope-intercept form, the slope formula and how to solve two-step equations.

Extra practice: The Algebra 1 Unit 4 Practice Pack (Part 1) has printable problems, answer keys and a cheat sheet for this unit.

What You Need to Write the Equation

Every line in slope-intercept form needs two numbers: the slope m and the y-intercept b. What you do depends on what you’re given.

What you know	What to do
The slope and the y-intercept	Put them straight into $y = mx + b$ .
The slope and one point	Substitute the slope and the point into $y = mx + b$ , then solve for b.
Two points	Find the slope with the slope formula first. Then use one point to solve for b.
A graph	Read b where the line crosses the y-axis. Count the rise and run to find m.

How to Find b From the Slope and a Point

Write $y = mx + b$ with your slope in place of m.
Put the point’s x-value in for x and its y-value in for y.
Solve the equation for b.
Write the final equation with your m and b.
Check: put the point back in and make sure both sides match.

Here is why this works. Every point on a line makes its equation true. So when you plug in a point, the only unknown left is b.

Worked Examples

Example 1: Slope and y-intercept

Write the equation of the line with slope −4 and y-intercept 7.

y = mx + b

Step 1. Write the slope-intercept form.

y = -4x + 7

Step 2. Substitute $m = -4$ and $b = 7$ .

Answer: $y = -4x + 7$

Example 2: Slope and one point

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

y = 3x + b

Step 1. Substitute the slope into $y = mx + b$ .

1 = 3(2) + b

Step 2. Substitute the point: $x = 2$ and $y = 1$ .

1 = 6 + b

Step 3. Multiply.

-5 = b

Step 4. Subtract 6 from both sides to solve for b (see one-step equations).

y = 3x - 5

Step 5. Write the equation with m = 3 and b = −5.

Answer: $y = 3x - 5$

Check: $3(2) - 5 = 6 - 5 = 1$ . The point (2, 1) is on the line.

Example 3: Two points

Write the equation of the line through (−2, 5) and (4, −1).

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

Step 1. Find the slope with the slope formula.

m = \frac{-1 - 5}{4 + 2}

Step 2. Subtracting −2 is the same as adding 2 (see adding and subtracting integers).

m = \frac{-6}{6}

Step 3. Do the top and the bottom.

m = -1

Step 4. Divide.

5 = -1(-2) + b

Step 5. Use the point (−2, 5) to find b. Substitute x = −2, y = 5 and m = −1.

5 = 2 + b

Step 6. Multiply. A negative times a negative is positive.

3 = b

Step 7. Subtract 2 from both sides.

y = -x + 3

Step 8. Write the equation with m = −1 and b = 3.

Answer: $y = -x + 3$

Check: Use the other point: $-(4) + 3 = -4 + 3 = -1$ . The point (4, −1) is on the line too.

Example 4: From a graph

Write the equation of the line in the graph.

Graph of a line that crosses the y-axis at (0, −2) and passes through (3, 2). A dashed path goes right 3 and up 4, so the slope is 4/3. — The line crosses the y-axis at (0, −2) and passes through (3, 2), so its slope is 4/3.

b = -2

Step 1. Find where the line crosses the y-axis: (0, −2).

Step 2. Find another point where the line crosses grid corners exactly: (3, 2).

2 - (-2) = 4

Step 3. Count the rise from (0, −2) up to the level of (3, 2). The line goes up 4.

3 - 0 = 3

Step 4. Count the run from x = 0 to x = 3. The line goes right 3.

m = \frac{4}{3}

Step 5. Divide the rise by the run.

y = \frac{4}{3}x - 2

Step 6. Substitute m and b into $y = mx + b$ .

Answer: $y = \frac{4}{3}x - 2$

Example 5: Word problem

A taxi charges a starting fee plus a cost per mile. A 3-mile ride costs $10.50, and an 8-mile ride costs $20.50. Write an equation for the cost C of a ride of d miles.

Step 1. Write the facts as points (miles, cost): (3, 10.50) and (8, 20.50).

m = \frac{20.50 - 10.50}{8 - 3}

Step 2. Find the slope, which is the cost per mile.

m = \frac{10}{5}

Step 3. Subtract on the top and on the bottom.

m = 2

Step 4. Divide.

10.50 = 2(3) + b

Step 5. Use (3, 10.50) to find b, the starting fee.

10.50 = 6 + b

Step 6. Multiply.

4.50 = b

Step 7. Subtract 6 from both sides.

Answer: $C = 2d + 4.50$ . The taxi charges $2 per mile plus a $4.50 starting fee.

Check: An 8-mile ride: $2(8) + 4.50 = 16 + 4.50 = 20.50$ . It matches.

Common Mistakes

Mistake	What it looks like	How to fix it
Using the point’s y-value as b	Slope 3 through (2, 1) written as $y = 3x + 1$	b is the y-value only when $x = 0$ . Solve for b.
Adding instead of subtracting	$5 = 2 + b$ , so $b = 7$	Subtract 2 from both sides: $b = 3$ .
Slope upside down	Points (1, 3) and (3, 11) give $m = \frac{2}{8}$	The y-values go on top: $m = \frac{8}{2} = 4$ .
Sign errors with negatives	$-1(-2) = -2$	A negative times a negative is positive: $-1(-2) = 2$ .
Skipping the check	Writing an equation that misses one of the points	Put the other point into your equation. Both sides should match.

Special Cases

Horizontal lines: The line through (−4, 3) and (6, 3) has slope $\frac{0}{10} = 0$ . Its equation is $y = 3$ .
Vertical lines: The line through (3, 1) and (3, 8) has the same x-value for both points. Its slope is undefined, so the equation is $x = 3$ . It can’t be written as $y = mx + b$ . See Horizontal and Vertical Lines.
Another method: Point-slope form, $y - y_1 = m(x - x_1)$ , is another way to write a line from a point and a slope. It has its own lesson later in this unit.

Practice Problems

Write each equation in slope-intercept form.

Slope 5, y-intercept −2
Slope $-\frac{2}{3}$ , y-intercept 4
Slope 2, through (3, 10)
Slope −3, through (−1, 8)
Slope $\frac{1}{4}$ , through (8, 1)
Through (1, 3) and (3, 11)
Through (−2, 6) and (2, −2)
Through (0, −5) and (5, 0)
Through (−4, 3) and (6, 3)
A plumber charges a fixed fee plus an hourly rate. A 2-hour job costs $190, and a 5-hour job costs $340. Write an equation for the cost C of an h-hour job.

Show answers

y = 5x - 2

Substitute m = 5 and b = −2 into y = mx + b.

y = -\frac{2}{3}x + 4

Substitute $m = -\frac{2}{3}$ and b = 4.

10 = 2(3) + b

Substitute m = 2 and the point (3, 10)

10 = 6 + b

Multiply.

b = 4

Subtract 6 from both sides.

The equation is $y = 2x + 4$ .

8 = -3(-1) + b

Substitute m = −3 and the point (−1, 8)

8 = 3 + b

Multiply.

b = 5

Subtract 3 from both sides.

The equation is $y = -3x + 5$ .

1 = \frac{1}{4}(8) + b

Substitute $m = \frac{1}{4}$ and the point (8, 1)

1 = 2 + b

Multiply.

b = -1

Subtract 2 from both sides.

The equation is $y = \frac{1}{4}x - 1$ .

m = \frac{11 - 3}{3 - 1} = \frac{8}{2} = 4

Find the slope.

3 = 4(1) + b

Substitute m = 4 and the point (1, 3)

3 = 4 + b

Multiply.

b = -1

Subtract 4 from both sides.

The equation is $y = 4x - 1$ .

m = \frac{-2 - 6}{2 - (-2)} = \frac{-8}{4} = -2

Find the slope.

6 = -2(-2) + b

Substitute m = −2 and the point (−2, 6)

6 = 4 + b

Multiply.

b = 2

Subtract 4 from both sides.

The equation is $y = -2x + 2$ .

m = \frac{0 - (-5)}{5 - 0} = \frac{5}{5} = 1

Find the slope.

b = -5

The point (0, −5) is on the y-axis, so it is the y-intercept.

The equation is $y = x - 5$ .

m = \frac{3 - 3}{6 - (-4)} = \frac{0}{10} = 0

Find the slope.

b = 3

A slope of 0 means a horizontal line. Both points have y = 3.

The equation is $y = 0x + 3$ , or $y = 3$ .

10.

Write the facts as points (hours, cost): (2, 190) and (5, 340)

m = \frac{340 - 190}{5 - 2} = \frac{150}{3} = 50

Find the slope.

190 = 50(2) + b

Substitute m = 50 and the point (2, 190)

190 = 100 + b

Multiply.

b = 90

Subtract 100 from both sides.

The equation is $C = 50h + 90$ : $50 per hour plus a $90 fee.

Want more practice? The Algebra 1 Unit 4 Practice Pack (Part 1) has 39 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Writing Equations FAQ

How do you write an equation in slope-intercept form from two points?

Find the slope with $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Then put the slope and one of the points into $y = mx + b$ and solve for b.

How do you find b in y = mx + b?

Put the slope and any point on the line into $y = mx + b$ , then solve for b. If you have a graph, b is where the line crosses the y-axis.

How do you write an equation from a graph?

Read the y-intercept where the line crosses the y-axis. Then count the rise and run between two grid points to find the slope.

Can you write a vertical line in slope-intercept form?

No. A vertical line has an undefined slope, so you write it as $x = a$ instead.

Summary

Slope-intercept form is $y = mx + b$ . You need m and b.
With two points, find the slope first.
To find b, put the slope and one point into $y = mx + b$ and solve.
On a graph, b is where the line crosses the y-axis.
Always check your equation with a second point.

Previous lesson: Slope-Intercept Form

Next lesson: Point-Slope Form

Next unit: Systems of Equations

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