Point-Slope Form

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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.

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

  1. Find the slope m. If you have two points, use the slope formula.
  2. Pick one point (x_1, y_1) on the line.
  3. Put m, x_1 and y_1 into y - y_1 = m(x - x_1).
  4. Clean up any double signs, like x - (-1), which becomes x + 1.
  5. 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.

Here m = 3, x_1 = 2 and y_1 = 5. Put them in the formula.

\begin{aligned} y - 5 &= 3(x - 2) \\ y - 5 &= 3x - 6 \\ y &= 3x - 1 \end{aligned}

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

\begin{aligned} y - 4 &= -2(x - (-1)) \\ y - 4 &= -2(x + 1) \\ y - 4 &= -2x - 2 \\ y &= -2x + 2 \end{aligned}

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).

First find the slope.

m = \frac{3 - (-3)}{4 - 1} = \frac{6}{3} = 2

Now use either point. With (1, -3):

\begin{aligned} y - (-3) &= 2(x - 1) \\ y + 3 &= 2x - 2 \\ y &= 2x - 5 \end{aligned}

Answer: y + 3 = 2(x - 1), or y = 2x - 5. Check the other point: 2(4) - 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).

Read the point and slope from the equation. Since x + 2 = x - (-2), the point is (-2, 1). The slope is -\tfrac{3}{4}.

  1. Plot (-2, 1).
  2. The slope -\tfrac{3}{4} means down 3 and right 4. From (-2, 1), that lands on (2, -2).
  3. 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.

Check: put (2, -2) into the equation. Left side: -2 - 1 = -3. Right side: -\tfrac{3}{4}(2 + 2) = -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?

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

\begin{aligned} y - 50 &= 12(x - 3) \\ y - 50 &= 12x - 36 \\ y &= 12x + 14 \end{aligned}

After 10 minutes: y = 12(10) + 14 = 134.

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

MistakeExample of the errorFix
Keeping the sign of a negative coordinatePoint (-4, 3): writing y - 3 = m(x - 4)x - (-4) is x + 4: y - 3 = m(x + 4).
Swapping the coordinatesPoint (2, 5): writing y - 2 = 3(x - 5)y_1 goes with y: y - 5 = 3(x - 2).
Distributing to only the first termy - 5 = 3(x - 2) becomes y - 5 = 3x - 2Multiply both terms: 3x - 6.
Forgetting to move y₁ when solving for yy - 5 = 3x - 6 becomes y = 3x - 6Add 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.

FormEquationBest when you know
Point-slope formy - y_1 = m(x - x_1)the slope and any point, or two points
Slope-intercept formy = mx + bthe slope and the y-intercept (0, b)
Standard formAx + By = Cyou 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

  1. Write the equation in point-slope form: slope 4 through (1, 3).
  2. Write the equation in point-slope form: slope -5 through (2, -1).
  3. Write the equation in point-slope form: slope \tfrac{2}{3} through (-3, 4).
  4. Name the point and the slope: y + 6 = -\tfrac{1}{2}(x - 8).
  5. Write y - 7 = 2(x + 3) in slope-intercept form.
  6. Write y + 4 = -3(x - 2) in slope-intercept form.
  7. Write the equation of the line through (2, 3) and (6, 11). Give both forms.
  8. Write the equation of the line through (-1, 5) and (3, -3). Give both forms.
  9. Write the equation of the line with slope 0 through (4, -2).
  10. 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
  1. y - 3 = 4(x - 1)
  2. y - (-1) = -5(x - 2), which is y + 1 = -5(x - 2).
  3. y - 4 = \tfrac{2}{3}(x - (-3)), which is y - 4 = \tfrac{2}{3}(x + 3).
  4. Since y + 6 = y - (-6), the point is (8, -6). The slope is -\tfrac{1}{2}.
  5. y - 7 = 2x + 6, so y = 2x + 13.
  6. y + 4 = -3x + 6, so y = -3x + 2.
  7. m = \tfrac{11 - 3}{6 - 2} = \tfrac{8}{4} = 2. Point-slope: y - 3 = 2(x - 2). Slope-intercept: y = 2x - 1.
  8. m = \tfrac{-3 - 5}{3 - (-1)} = \tfrac{-8}{4} = -2. Point-slope: y - 5 = -2(x + 1). Slope-intercept: y = -2x + 3.
  9. y + 2 = 0(x - 4), so y + 2 = 0 and y = -2. This is a horizontal line.
  10. The slope is 25 and the point is (5, 185). y - 185 = 25(x - 5), so y - 185 = 25x - 125 and y = 25x + 60. The sign-up fee is $60.

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

Back to the Algebra 1 Index

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