Lesson 7: Linear Equations and Intersections

Learning Objectives

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

  • Write equations of lines in slope-intercept and point-slope forms
  • Convert between equation forms
  • Find the intersection point of two lines

Equations of Lines

Slope-Intercept Form

\[y = mx + c\]

Point-Slope Form

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

General Form

\[ax + by + c = 0\]

Example 1: Equation from Gradient and Point

Write the equation of the line with gradient 3 passing through (2, 5).

Solution: \[y - 5 = 3(x - 2)\] \[y - 5 = 3x - 6\] \[y = 3x - 1\]

Example 2: Equation from Two Points

Find the equation of the line through P(1, 3) and Q(4, 9).

Solution: \[m = \frac{9 - 3}{4 - 1} = 2\] \[y - 3 = 2(x - 1)\] \[y = 2x + 1\]

Intersection of Two Lines

The intersection point solves both equations at the same time.

Example 3: Intersection by Substitution

Find the intersection of:

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

Solution: Set them equal: \[2x + 1 = -x + 7\] \[3x = 6\] \[x = 2\] Substitute back: \[y = 2(2) + 1 = 5\]

Intersection is \((2, 5)\).

Example 4: Intersection with General Form

Find the intersection of:

\[2x + y - 8 = 0\] \[x - y + 1 = 0\]

Solution: From the second equation: \(y = x + 1\). Substitute into the first: \[2x + (x + 1) - 8 = 0\] \[3x - 7 = 0\] \[x = \frac{7}{3}\] Then \(y = \frac{7}{3} + 1 = \frac{10}{3}\).

Intersection is \(\left(\frac{7}{3}, \frac{10}{3}\right)\).

Practice Exercises

  1. Write the equation of the line:
    • With gradient -2 and y-intercept 5
    • Through point (3, 7) with gradient -1
    • Through points (0, 2) and (5, 12)
  2. Convert to slope-intercept form:
    • \(3x + y - 6 = 0\)
    • \(2x - 4y + 8 = 0\)
  3. Find the intersection point of each pair:
    • \(y = x + 3\) and \(y = -2x + 9\)
    • \(y = 4x - 5\) and \(2x + y = 7\)

Worksheet 2B: Linear Equations and Intersections

  1. Write the equation of the line:
    • With gradient 3 and y-intercept 5
    • With gradient -2 passing through (4, 1)
    • Through points (2, 3) and (6, 11)
  2. Convert to slope-intercept form (\(y = mx + c\)):
    • \(x + 2y - 10 = 0\)
    • \(4x - 2y + 6 = 0\)
  3. Find the intersection point:
    • \(y = 2x - 1\) and \(y = -x + 8\)
    • \(3x + y = 12\) and \(x - y = 4\)
  1. \(y = 3x + 5\), \(y = -2x + 9\), \(y = 2x - 1\)
  2. \(y = -\frac{1}{2}x + 5\), \(y = 2x + 3\)
  3. (3, 5), (4, 0)

Previous Lesson: Lesson 6: Gradient Using the Formula
Next Lesson: Lesson 8: Parallel Lines