Lesson 10: Equation of a Line from Gradient and a Point

Learning Objectives

By the end of this lesson, students will be able to:

  • Use point-slope form to write an equation from a gradient and a point
  • Convert equations to slope-intercept form
  • Check whether a point lies on a line

Point-Slope Form

If a line has gradient \(m\) and passes through \((x_1, y_1)\), then:

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

Example 1: Write an Equation from a Gradient and a Point

Find the equation of the line with gradient 4 passing through (2, -1).

Solution: \[y - (-1) = 4(x - 2)\] \[y + 1 = 4x - 8\] \[y = 4x - 9\]

Example 2: Keep in Point-Slope Form

Find the equation of the line with gradient \(-\frac{3}{2}\) passing through (6, 5).

Solution: \[y - 5 = -\frac{3}{2}(x - 6)\]

Example 3: Check a Point

Does (3, 7) lie on the line \(y = 2x + 1\)?

Solution: \[2(3) + 1 = 7\] Yes, the point lies on the line.

Practice Exercises

  1. Write the equation of the line with gradient 3 passing through (-1, 4).
  2. Write the equation of the line with gradient \(-2\) passing through (5, 9).
  3. Convert your equations from (1) and (2) to slope-intercept form.
  4. Determine whether (4, -3) lies on the line \(y = -x + 1\).

Worksheet 2C: Gradient and a Point

  1. Write the equation of the line:
    • With gradient 2 passing through (1, -3)
    • With gradient \(-\frac{1}{2}\) passing through (6, 4)
  2. Convert to slope-intercept form:
    • \(y - 7 = 5(x + 1)\)
    • \(y + 2 = -3(x - 4)\)
  3. Check whether each point lies on the line \(y = 3x - 2\):
    • (1, 1)
    • (2, 4)
  1. \(y = 2x - 5\), \(y = -\frac{1}{2}x + 7\)
  2. \(y = 5x + 12\), \(y = -3x + 10\)
  3. Yes, yes

Previous Lesson: Lesson 9: Perpendicular Lines
Next Lesson: Lesson 11: Practice and Revision