Simultaneous equations graphically

Solving a pair of linear equations by graphing both lines and reading the point of intersection, checking the answer, and recognising parallel and identical lines.

Two equations, two unknowns, one answer that has to work in both at once. That is what "simultaneous" means. A graph shows it beautifully: each equation is a line, and the answer is the one point that sits on both.

Revise: gradient, intercept and $y = mx + c$ are in Year 11 Linear relationships.

Why the crossing point is the answer

Every point on the line $y = 2x + 1$ makes that equation true. Every point on $y = -x + 7$ makes the second equation true. A point on both lines makes both true at the same time, and two different straight lines with different gradients meet at exactly one point.

Solving graphically

1. Graph both lines on the same axes.
2. Read the coordinates of the point of intersection.
3. Check by substituting into both equations.

Example 1●

Solve $y = 2x + 1$ and $y = -x + 7$ graphically.

$y = 2x + 1$ passes through $(0, 1)$ and $(3, 7)$. $y = -x + 7$ passes through $(0, 7)$ and $(7, 0)$.

Two points are enough for a straight line. The intercepts or a small table are quickest.

The lines cross at $(2, 5)$.

Plot both on one set of axes and look for the crossing.

$2 \times 2 + 1 = 5$ ✓ and $-2 + 7 = 5$ ✓, so $x = 2$, $y = 5$.

A graph can be misread by a little, so always substitute back.

InteractiveWhere do the lines meet?

Try this

Start on Crossing and read the solution. Then change Line 2's intercept until the lines meet on the y-axis. What do you notice about the two intercepts? Next pick Parallel and move one intercept: do the lines ever meet? Finally make the gradients $1$ and $-0.5$ with intercepts $0$ and $2$, and notice the crossing is not on a grid point.

Equations not in y = mx + c form

Many questions give equations like $2x + 3y = 12$. You can rearrange to $y = mx + c$, but the fastest way to graph them is with the two intercepts: put $x = 0$ to find where it cuts the $y$-axis and $y = 0$ for the $x$-axis.

Example 2●●

Solve $x + y = 6$ and $x - 2y = 0$ graphically.

$x + y = 6$ passes through $(0, 6)$ and $(6, 0)$.

Intercepts for the first line: x = 0 gives y = 6, and y = 0 gives x = 6.

$x - 2y = 0$ passes through $(0, 0)$ and $(6, 3)$.

The second line passes through the origin, so both intercepts are 0 and you need another point. Choose x = 6, which makes 6 − 2y = 0.

The lines meet at $(4, 2)$.

Read the crossing from the graph.

$4 + 2 = 6$ ✓ and $4 - 2 \times 2 = 0$ ✓

Check in both.

No solution or infinitely many

Parallel lines have the same gradient but different intercepts. They never meet, so there is no solution. Example: $y = 2x - 3$ and $y = 2x + 1$.
Identical lines are the same equation in disguise, such as $y = -x + 3$ and $2x + 2y = 6$. Every point on the line works, so there are infinitely many solutions.

Example 3●●

How many solutions does the pair $y = 3x - 2$ and $6x - 2y = 10$ have?

$-2y = -6x + 10$, so $y = 3x - 5$.

Put the second equation in y = mx + c form so the gradients can be compared.

The lines are parallel, so there is no solution.

Same gradient 3, different intercepts (−2 and −5).

Using a graphing app

When the crossing is not on grid lines, a hand-drawn graph only gives an estimate. A graphing app such as Desmos or GeoGebra will give the exact intersection: type both equations and tap the point where they cross. In an exam without an app, give a graphical answer to the accuracy of the grid and say it is an estimate, or solve algebraically (next section).

Your turn

Solve $y = x - 1$ and $y = -2x + 8$ graphically, and check your answer.

Show the working

Points for the first line: $(0, -1)$ and $(4, 3)$. Points for the second: $(0, 8)$ and $(4, 0)$. The lines cross at $(3, 2)$. Check: $3 - 1 = 2$ ✓ and $-6 + 8 = 2$ ✓

Flashcards

Graphical solution

The point of intersection: it lies on both lines, so it satisfies both equations

Parallel lines

Same gradient, different intercept: no solution

Identical lines

Same line: infinitely many solutions

Practice

Why does the point where two lines cross solve both equations?
The lines $y = 2x + 1$ and $y = -x + 7$ cross at which point? Give it as x, y.
Where does $x + 2y = 8$ cut the $y$-axis? Give the $y$-value.
How many solutions do $y = 4x + 1$ and $y = 4x - 6$ have?
How many solutions do $y = -x + 3$ and $2x + 2y = 6$ have?
Is $(1, 4)$ the solution of $y = 3x + 1$ and $y = x + 2$?
Solve $y = x - 1$ and $y = -2x + 8$. Give x, y.
Solve $x + y = 6$ and $x - 2y = 0$. Give x, y.
Exam style: Two lines on a graph cross between grid lines, at about $(1.3, 2.7)$. What is the best way to state the answer?
Exam style: Two water tanks are draining. After $t$ minutes, tank A holds $V = 400 - 30t$ litres and tank B holds $V = 310 - 12t$ litres. After how many minutes do they hold the same volume?