Point-Slope Form: Formula and Examples

Point-slope form is a way to write the equation of a line when you know its slope and one point on it. The form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point. For slope 3 through $(2, 5)$ , the equation is $y - 5 = 3(x - 2)$ .

Before this lesson: You should know how to find slope, use slope-intercept form, and use the distributive property.

Extra practice: The Algebra 1 Unit 4 Practice Pack (Part 2) has printable problems, answer keys and a cheat sheet for this unit.

What Is Point-Slope Form?

A line is set once you know two things: how steep it is and one place it passes through. Point-slope form uses exactly those two facts.

Here is where it comes from. Pick a fixed point $(x_1, y_1)$ on the line. Let $(x, y)$ be any other point on the line. The slope between them is the rise over the run.

m = \frac{y - y_1}{x - x_1}

Multiply both sides by $x - x_1$ . You get point-slope form. Every point $(x, y)$ on the line makes it true.

A line with slope 2 on a grid from −3 to 5. A fixed point (x₁, y₁) = (1, −1) and another point (x, y) on the line are marked. A dashed green right triangle between them shows the run, x − x₁, across the bottom and the rise, y − y₁, up the side. Text beside the graph reads: slope = rise ÷ run, so m = (y − y₁) ÷ (x − x₁), then multiply both sides to get y − y₁ = m(x − x₁). — The slope between a fixed point and any other point on the line is m. Multiplying both sides gives point-slope form.

The Point-Slope Form Formula

y - y_1 = m(x - x_1)

$m$ is the slope.
$(x_1, y_1)$ is one known point on the line.
$x$ and $y$ stay as letters. They stand for every point on the line.

The formula has minus signs built in. So a negative coordinate turns into a plus sign. For the point $(-4, 3)$ , you get $y - 3 = m(x + 4)$ .

Steps to write an equation in point-slope form

Find the slope $m$ . If you have two points, use the slope formula.
Pick one point $(x_1, y_1)$ on the line.
Put $m$ , $x_1$ and $y_1$ into $y - y_1 = m(x - x_1)$ .
Clean up any double signs, like $x - (-1)$ , which becomes $x + 1$ .
If asked, distribute and solve for $y$ to get slope-intercept form.

Worked Examples

Example 1: Slope and a point

Write the equation of the line with slope 3 through $(2, 5)$ . Then write it in slope-intercept form.

y - 5 = 3(x - 2)

Step 1. Name the parts: $m = 3$ , $x_1 = 2$ and $y_1 = 5$ . Put them in the formula.

y - 5 = 3x - 6

Step 2. Use the distributive property on the right side.

y = 3x - 1

Step 3. Add 5 to both sides.

Answer: $y - 5 = 3(x - 2)$ , or $y = 3x - 1$ .

Check: When $x = 2$ , $y = 3(2) - 1 = 6 - 1 = 5$ . It works.

Example 2: A negative slope and a negative coordinate

Write the equation of the line with slope $-2$ through $(-1, 4)$ .

y - 4 = -2(x - (-1))

Step 1. Name the parts: $m = -2$ , $x_1 = -1$ and $y_1 = 4$ . Put them in the formula.

y - 4 = -2(x + 1)

Step 2. Subtracting −1 is the same as adding 1 (see adding and subtracting integers).

y - 4 = -2x - 2

Step 3. Distribute −2 to both terms in the parentheses.

y = -2x + 2

Step 4. Add 4 to both sides.

Answer: $y - 4 = -2(x + 1)$ , or $y = -2x + 2$ . Notice how $x - (-1)$ became $x + 1$ .

Example 3: Two points

Write the equation of the line through $(1, -3)$ and $(4, 3)$ .

m = \frac{3 - (-3)}{4 - 1}

Step 1. Find the slope with the slope formula.

m = \frac{6}{3} = 2

Step 2. Simplify the top and bottom, then divide.

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

Step 3. Use either point. With $(1, -3)$ , put m = 2, x₁ = 1 and y₁ = −3 in the formula.

y + 3 = 2(x - 1)

Step 4. Subtracting −3 is the same as adding 3.

y + 3 = 2x - 2

Step 5. Distribute the 2.

y = 2x - 5

Step 6. Subtract 3 from both sides.

Answer: $y + 3 = 2(x - 1)$ , or $y = 2x - 5$ .

Check: Use the other point: $2(4) - 5 = 8 - 5 = 3$ . It works. If you use $(4, 3)$ instead, you get $y - 3 = 2(x - 4)$ . That looks different, but it also simplifies to $y = 2x - 5$ . It is the same line.

Example 4: Graph a line from point-slope form

Graph $y - 1 = -\tfrac{3}{4}(x + 2)$ .

x + 2 = x - (-2)

Step 1. Rewrite the inside so it matches $y - y_1 = m(x - x_1)$ .

(-2, 1)

Step 2. Read the point and plot it.

-\tfrac{3}{4}

Step 3. Read the slope. It means down 3 and right 4.

1 - 3 = -2

Step 4. From (−2, 1), go down 3 to find the new y-value.

-2 + 4 = 2

Step 5. Then go right 4 to find the new x-value. Plot (2, −2).

Step 6. Draw a line through the two points.

The line y − 1 = −3/4(x + 2) on a grid from −5 to 5. The starting point (−2, 1) is marked with an orange dot. From it, a dashed green path goes 4 units right and 3 units down to a second point, (2, −2). The line passes through both points. Text beside the graph: point (−2, 1); slope −3/4 means down 3, right 4. — Start at the point (−2, 1). The slope −3/4 means go down 3 and right 4.

Answer: The line through $(-2, 1)$ and $(2, -2)$ .

Check: Put $(2, -2)$ into the equation. Left side: $-2 - 1 = -3$ . Right side: $-\tfrac{3}{4}(2 + 2) = -\tfrac{3}{4}(4) = -3$ . Both sides match.

Example 5: Word problem with a water tank

A tank fills at a steady 12 gallons per minute. After 3 minutes, it holds 50 gallons. Write an equation for the water $y$ after $x$ minutes. How much water is in the tank after 10 minutes?

Step 1. The rate is the slope, so $m = 12$ . The fact “after 3 minutes, 50 gallons” is the point $(3, 50)$ .

y - 50 = 12(x - 3)

Step 2. Put them in the formula.

y - 50 = 12x - 36

Step 3. Distribute the 12.

y = 12x + 14

Step 4. Add 50 to both sides.

y = 12(10) + 14

Step 5. Substitute $x = 10$ .

y = 120 + 14 = 134

Step 6. Multiply, then add.

Answer: $y = 12x + 14$ . The tank holds 134 gallons after 10 minutes. The 14 also tells you the tank started with 14 gallons.

Common Mistakes

Mistake	Example of the error	Fix
Keeping the sign of a negative coordinate	Point $(-4, 3)$ : writing $y - 3 = m(x - 4)$	$x - (-4)$ is $x + 4$ : $y - 3 = m(x + 4)$ .
Swapping the coordinates	Point $(2, 5)$ : writing $y - 2 = 3(x - 5)$	$y_1$ goes with $y$ : $y - 5 = 3(x - 2)$ .
Distributing to only the first term	$y - 5 = 3(x - 2)$ becomes $y - 5 = 3x - 2$	Multiply both terms: $3x - 6$ .
Forgetting to move y₁ when solving for y	$y - 5 = 3x - 6$ becomes $y = 3x - 6$	Add 5 to both sides: $y = 3x - 1$ .

Point-Slope Form vs. Other Forms

A line can be written in more than one form. Each form is handy for a different job.

Form	Equation	Best when you know
Point-slope form	$y - y_1 = m(x - x_1)$	the slope and any point, or two points
Slope-intercept form	$y = mx + b$	the slope and the y-intercept $(0, b)$
Standard form	$Ax + By = C$	you want to find both intercepts fast

Point-slope form is often the fastest way to start. Then you can change it to slope-intercept form by distributing and solving for $y$ . To graph with x- and y-intercepts, standard form works best. The next lesson covers standard form.

One more note: a horizontal line has slope 0. In point-slope form, $y + 2 = 0(x - 4)$ becomes $y = -2$ . A vertical line has no slope, so it can’t be written in point-slope form. See horizontal and vertical lines.

Practice Problems

Write the equation in point-slope form: slope 4 through $(1, 3)$ .
Write the equation in point-slope form: slope $-5$ through $(2, -1)$ .
Write the equation in point-slope form: slope $\tfrac{2}{3}$ through $(-3, 4)$ .
Name the point and the slope: $y + 6 = -\tfrac{1}{2}(x - 8)$ .
Write $y - 7 = 2(x + 3)$ in slope-intercept form.
Write $y + 4 = -3(x - 2)$ in slope-intercept form.
Write the equation of the line through $(2, 3)$ and $(6, 11)$ . Give both forms.
Write the equation of the line through $(-1, 5)$ and $(3, -3)$ . Give both forms.
Write the equation of the line with slope 0 through $(4, -2)$ .
A gym charges a sign-up fee plus $25 per month. After 5 months, Leo has paid $185 in all. Write an equation for the total cost $y$ after $x$ months. What is the sign-up fee?

Show answers

Name the parts: m = 4, x₁ = 1, y₁ = 3

y - 3 = 4(x - 1)

Put them in the formula.

Name the parts: m = −5, x₁ = 2, y₁ = −1

y - (-1) = -5(x - 2)

Put them in the formula.

y + 1 = -5(x - 2)

Subtracting −1 is adding 1.

Name the parts: m = 2/3, x₁ = −3, y₁ = 4

y - 4 = \tfrac{2}{3}(x - (-3))

Put them in the formula.

y - 4 = \tfrac{2}{3}(x + 3)

Subtracting −3 is adding 3.

y + 6 = y - (-6)

Rewrite the left side as y minus something.

(8, -6)

Read the point: x₁ = 8 and y₁ = −6.

m = -\tfrac{1}{2}

Read the number in front of the parentheses.

y - 7 = 2x + 6

Distribute the 2.

y = 2x + 13

Add 7 to both sides.

y + 4 = -3x + 6

Distribute the −3. A negative times a negative is positive.

y = -3x + 2

Subtract 4 from both sides.

m = \tfrac{11 - 3}{6 - 2} = \tfrac{8}{4} = 2

Find the slope.

y - 3 = 2(x - 2)

Use the point (2, 3)

y - 3 = 2x - 4

Distribute the 2.

y = 2x - 1

Add 3 to both sides.

y - 3 = 2(x - 2)

Point-slope. Slope-intercept: $y = 2x - 1$ .

m = \tfrac{-3 - 5}{3 - (-1)} = \tfrac{-8}{4} = -2

Find the slope.

y - 5 = -2(x - (-1))

Use the point (−1, 5)

y - 5 = -2(x + 1)

Subtracting −1 is adding 1.

y - 5 = -2x - 2

Distribute the −2.

y = -2x + 3

Add 5 to both sides.

y - 5 = -2(x + 1)

Point-slope. Slope-intercept: $y = -2x + 3$ .

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

Put m = 0 and the point (4, −2) in the formula.

y + 2 = 0

Simplify both sides. Zero times anything is 0.

y = -2

Subtract 2 from both sides.

This is a horizontal line.

10.

y - 185 = 25(x - 5)

The slope is 25, and the point is (5, 185)

y - 185 = 25x - 125

Distribute the 25.

y = 25x + 60

Add 185 to both sides.

The sign-up fee is the y-intercept, $60.

Want more practice? The Algebra 1 Unit 4 Practice Pack (Part 2) has 38 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Point-Slope Form FAQ

What is the point-slope form of a line?

It is $y - y_1 = m(x - x_1)$ . You use it to write a line’s equation from its slope $m$ and one point $(x_1, y_1)$ .

How do you change point-slope form to slope-intercept form?

Distribute the slope on the right side. Then add or subtract to get $y$ alone. For example, $y - 5 = 3(x - 2)$ becomes $y = 3x - 1$ .

When should you use point-slope form?

Use it when you know a point and the slope, or when you know two points. It saves you from finding the y-intercept first.

Does it matter which point you use?

No. Any point on the line works. The point-slope equations look different, but they simplify to the same slope-intercept equation.

Summary

Point-slope form is $y - y_1 = m(x - x_1)$ .
$m$ is the slope and $(x_1, y_1)$ is any known point on the line.
A negative coordinate flips to a plus sign: the point $(-1, 4)$ gives $y - 4 = m(x + 1)$ .
With two points, find the slope first, then use either point.
Distribute and solve for $y$ to get slope-intercept form.

Previous lesson: Writing Equations in Slope-Intercept Form

Next lesson: Standard Form of a Linear Equation

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