Solving Simultaneous Equations Graphically

What "simultaneous" means

Simultaneous equations are two (or more) equations that are true at the same time. For example:

y = x + 1
y = −x + 5

A solution is a pair of values — one for x, one for y — that satisfies both equations together. The graphical method finds that pair by drawing each equation as a straight line and seeing where they meet.

If you have already met the algebra route, this lesson complements simultaneous equations by showing what the answer looks like.

The key idea

Each equation, when plotted, gives a straight line. Every point on a line represents an (x, y) pair that makes that equation true.

The point where the two lines cross lies on both lines, so its coordinates make both equations true. That crossing point is the solution.

How to plot a line quickly

To draw a line such as y = x + 1, build a small table of values. Pick a few x-values, work out each y, and plot the points. For more detail, see straight-line graphs and gradients.

x	y = x + 1
0	1
2	3
4	5

Worked example 1

Solve graphically: y = x + 1 and y = −x + 5.

Step 1 — table for the first line y = x + 1:

x	0	2	4
y	1	3	5

Step 2 — table for the second line y = −x + 5:

x	0	2	5
y	5	3	0

Step 3 — plot both lines on the same axes and find where they cross. Both tables contain the point (2, 3), and that is where the lines intersect.

Step 4 — read the solution: x = 2, y = 3.

Step 5 — check in both equations:

• y = x + 1 → 3 = 2 + 1 ✓
• y = −x + 5 → 3 = −2 + 5 ✓

Worked example 2: rearrange first

Sometimes an equation is not in the form y = .... Rearrange it before plotting.

Solve graphically: y = 2x − 1 and x + y = 5.

Rearrange the second: y = 5 − x.

x	y = 2x − 1	y = 5 − x
0	−1	5
1	1	4
2	3	3
3	5	2

Both lines pass through (2, 3), so the solution is x = 2, y = 3.

Check: y = 2x − 1 → 3 = 2(2) − 1 = 3 ✓ and x + y = 5 → 2 + 3 = 5 ✓.

Worked example 3: no solution (parallel lines)

Plot y = x + 1 and y = x + 4.

x	y = x + 1	y = x + 4
0	1	4
2	3	6

Both lines have the same gradient (the number in front of x is 1), so they rise at the same rate and stay 3 apart forever. They are parallel and never cross, so there is no solution.

Worked example 4: infinitely many solutions (same line)

Plot y = 2x + 1 and 2y = 4x + 2.

Divide the second by 2: y = 2x + 1 — exactly the same equation. Both draw the identical line, so every point on it satisfies both. There are infinitely many solutions.

Summary of the three cases

Picture	Number of solutions
Lines cross once	exactly one
Lines are parallel	none
Lines are identical	infinitely many

A reliable method

1. Rearrange each equation into the form y = ... if needed.
2. Make a small table of values for each and plot the line.
3. Find the point where the lines cross.
4. Read off its x and y — that is your solution.
5. Check by substituting into both original equations.

Activity: solve graphically

For each pair, build short tables, plot, and state the solution (or say "no solution" / "infinitely many").

1. y = x and y = 4 − x
2. y = x + 2 and y = 2x
3. y = x − 3 and y = x + 1
4. y = 3x and y = x + 4

Answers:

1. Cross at (2, 2) → x = 2, y = 2.
2. Cross at (2, 4) → x = 2, y = 4. (At x = 2 both give y = 4.)
3. Same gradient, different intercepts → parallel → no solution.
4. Cross at (2, 6) → x = 2, y = 6. (3x = x + 4 → 2x = 4 → x = 2.)

Where this leads

Seeing solutions as crossing points connects algebra to geometry. The same "where do graphs meet?" idea reappears with curves: where a line crosses a curve, or where a quadratic equations graph meets the x-axis, are solutions too.