Point-Slope Form

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Use point-slope form to write equations of lines.
  • Convert point-slope equations into slope-intercept form.
  • Write equations when given a point and slope, or two points.

Key Ideas

Point-slope form is incredibly useful when a graph or problem gives you one point and the slope, or when you need to build an equation from two points.

Point-slope form:

\[ y - y_1 = m(x - x_1) \]

Where:

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

You can rewrite it into slope-intercept form by:

  1. Distributing the slope
  2. Moving \(y_1\) to the right side

Common Problem Types

1. Write an equation using a point and a slope

Use the given point \((x_1, y_1)\) and slope \(m\) directly in
\(y - y_1 = m(x - x_1)\).

2. Convert point-slope to slope-intercept

Distribute and simplify to get \(y = mx + b\).

3. Write an equation from two points

Step 1: Compute slope
Step 2: Use point-slope form
Step 3: Optionally convert to slope-intercept form

Strategies

  • Always place the point exactly as written: \(y - y_1\) and \(x - x_1\).
  • When the point has negatives, keep track of double negatives:
    • For example, \((x - (-3))\) becomes \((x + 3)\).
  • After using point-slope, check your equation by plugging in the given point.
  • If an equation looks messy, convert to slope-intercept to clean it up.

Worked Examples

Example 1 — Use point and slope

Write the equation of a line with slope \(m = 2\) through \((3, 7)\).

Use point-slope:

\[ y - 7 = 2(x - 3) \]

Convert:

\[ y - 7 = 2x - 6 \Rightarrow y = 2x + 1 \]

Example 2 — Use two points

Points: \((1, -2)\) and \((5, 6)\)

Step 1: Find slope

\[ m = \frac{6 - (-2)}{5 - 1} = \frac{8}{4} = 2 \]

Step 2: Use point-slope with point \((1, -2)\)

\[ y + 2 = 2(x - 1) \]

Step 3: Convert (optional)

\[ y + 2 = 2x - 2 \Rightarrow y = 2x - 4 \]

WarningCommon Mistakes
  • Forgetting parentheses when writing \(x - x_1\).
  • Using the wrong sign for \(x_1\) or \(y_1\).
  • Distributing incorrectly when converting.
  • Confusing point-slope with slope-intercept.

Practice Problems

  1. Write the equation of a line with slope \(3\) through \((2, 5)\).
  2. Convert to slope-intercept:
    \(y - 4 = -2(x + 1)\).
  3. Write an equation through \((0, -3)\) and \((4, 1)\).
  4. Write point-slope form for slope \(-5\) through \((-1, 7)\).
  5. Use point-slope to write an equation that passes through \((6, -2)\) and has slope \(\frac{1}{3}\).

1.
\[ y - 5 = 3(x - 2) \]
Convert (optional):
\[ y = 3x - 1 \]

2.
\[ y - 4 = -2(x + 1) = -2x - 2 \Rightarrow y = -2x + 2 \]

3.
Slope:
\[ m = \frac{1 - (-3)}{4 - 0} = 1 \]
Equation:
\[ y + 3 = 1(x - 0) \Rightarrow y = x - 3 \]

4.
\[ y - 7 = -5(x + 1) \]

5.
\[ y + 2 = \frac{1}{3}(x - 6) \]

Summary

  • Point-slope form writes a line using a point and slope.
  • Convert to slope-intercept by distributing and isolating \(y\).
  • Works especially well when a graph or problem gives you a point but not the intercept.
  • If a graph doesn’t show the intercept clearly, switch to point-slope.
  • When the point has negatives, rewrite carefully to avoid sign errors.
  • Plug the original point into your final equation for a quick accuracy check.