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Maths

Solving Simultaneous Equations Graphically

Overview

Before the algebraic methods, students develop a visual understanding of simultaneous equations — what it means for two equations to both be true, and why the solution is a point on both lines.

Learning Objective
Students solve pairs of simultaneous linear equations by plotting both lines on the same axes and reading off the coordinates of the intersection point, understanding what this solution represents.

Resources needed

  • Graph paper or coordinate grids
  • Ruler
  • Pencil
  • Calculator (optional)

Lesson stages

0 / 7 done
  1. 1 Equation 1: y = 2x + 1. Ask: 'Give me a pair (x,y) that satisfies this.' (0,1), (1,3), (2,5) — infinitely many. 'Now I also need y = −x + 7. Give me a pair satisfying this.' 'Can you find ONE pair that satisfies BOTH?' That is a simultaneous solution.
  2. 2 Make a table of values for y = 2x+1 (x = 0,1,2,3) and for y = −x+7 (x = 0,1,2,3). Plot both on the same axes. 'Where do they cross?' Read off: (2, 5). Verify: 2(2)+1=5 ✓ and −2+7=5 ✓.
  3. 3 'Every point on y=2x+1 satisfies the first equation. Every point on y=−x+7 satisfies the second. Only ONE point satisfies both — the intersection.' Students explain this to a partner in their own words.
  4. 4 Rearrange x+2y=8 and 3x−y=5 into y=mx+c form before plotting. y=(8−x)/2=−½x+4 and y=3x−5. Plot and find the intersection. Verify algebraically.
  5. 5 What happens if the lines are parallel (same gradient)? Plot y=2x+3 and y=2x−1. No intersection — no solution. What if they are the same line? Infinitely many solutions. Students identify these from the equations before plotting.
  6. 6 Do all three lines meet at a point? y=x+2, y=−x+4, y=2x−1. Plot all three. Do they all pass through the same point? (If not — no single solution for all three simultaneously.)
  7. 7 Students write: 'The solution to a pair of simultaneous equations is ___. Graphically, this means ___. If there is no solution, the lines are ___.'

Tap a step to mark it as done.

Variations

  • Compare graphical and algebraic methods for accuracy and efficiency
  • Include a quadratic and linear equation (two intersection points)
  • Use technology (Desmos) to check graphical solutions quickly
More information

Display: simultaneous, intersection, satisfy, solution, verify. Use: 'The point (___, ___) satisfies y=___ because ___.'

Provide pre-drawn axes. Give a table of values for one of the equations. Restrict to equations already in y=mx+c form.

Are lines plotted accurately (at least 3 points each)? Is the intersection read to the correct precision? Do students verify the solution algebraically?

Draw axes on plain paper. Restrict to integer intersection points initially for accuracy.

Students may read the intersection inaccurately if lines cross at non-integer coordinates. Discuss: algebraic methods give exact answers when graphs are imprecise.

Activity details
Secondary Medium 40 minutes 5 minutes prep Solo, Pairs Indoor
Topics
Simultaneous Equations Graphs Algebra Linear Equations