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Point-Slope Form

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

Point-slope form uses a known point and a slope to write the equation of a line: y - y1 = m(x - x1). It is especially useful when you know a point on the line but do not immediately know the y-intercept. The point gives the anchor, and the slope tells how the line moves from that anchor. When practicing, substitute the given point for x1 and y1, and substitute the slope for m. Be careful with signs: subtracting a negative coordinate becomes addition. A common mistake is switching x1 and y1 or losing the negative sign inside the parentheses.

What you'll learn

Why it matters: Point-slope form is handy when you know one measured data point and a rate, such as a starting temperature and a cooling rate.

Worked example

Problem. Write a line with slope 3 through (2, 5).

  1. Use y - y1 = m(x - x1).
  2. Substitute m = 3, x1 = 2, and y1 = 5.
  3. The equation is y - 5 = 3(x - 2).

Answer: y - 5 = 3(x - 2)

Practice problems

1. Which is point-slope form?

Choices: y - 4 = 2(x - 1) · y = 2x + 4 · 2x + y = 4 · x = 4

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which is point-slope form?
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Point-slope form shows y - y1 and x - x1.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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

2. Write a line with slope 2 through (3, 7).

Choices: y - 7 = 2(x - 3) · y - 3 = 2(x - 7) · y = 2x + 7 · y + 7 = 2(x + 3)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Write a line with slope 2 through (3, 7).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Use x1 = 3 and y1 = 7.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y - 7 = 2(x - 3)

3. In y - 5 = 4(x - 2), what is the slope?

Show solution
  1. Warm-up: First identify exactly what the question is asking: In y - 5 = 4(x - 2), what is the slope?
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. The number multiplying the parentheses is m.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. Write a line with slope -1 through (4, 6).

Choices: y - 6 = -1(x - 4) · y - 4 = -1(x - 6) · y = -x + 6 · y + 6 = -1(x + 4)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Write a line with slope -1 through (4, 6).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Substitute m = -1, x1 = 4, y1 = 6.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y - 6 = -1(x - 4)

5. Write a line with slope 3 through (-2, 5).

Choices: y - 5 = 3(x + 2) · y + 5 = 3(x - 2) · y - 2 = 3(x + 5) · y = 3x - 2

Show solution
  1. Core Practice: First identify exactly what the question is asking: Write a line with slope 3 through (-2, 5).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. x - (-2) becomes x + 2.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y - 5 = 3(x + 2)

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