What is the solution of a system of equations?

A pair (x, y) that makes both equations true at once. On a graph it is the point where the two lines cross.

How do you solve a system of equations by graphing?

Graph both lines on the same axes, read the point where they cross, and check that the point works in both equations.

What if the lines do not cross on a grid point?

Then graphing only gives an estimate. Use substitution or elimination to find the exact answer.

Can a system of equations have no solution?

Yes. If the lines are parallel they never cross, so no point works in both equations. If they are the same line, every point on it is a solution.

Part 15 of 20 · EquationsSee the whole course

How to solve a system of equations by graphing

By The Pip team · October 23, 2026 · 6 min read

Short answer

Each equation in a system is a line. Graph both lines on the same axes. The solution is the point where they cross, because it is the only point that sits on both lines and so works in both equations.

What a system of equations is

A system is two equations that share the same unknowns, x and y. For example:

y = 2x − 2
y = −x + 7

A solution is a pair (x, y) that makes both equations true at once. Each line is a picture of all the pairs that make one equation true. A pair on both lines works for both.

The method in four steps

Graph the first line. Put a dot at b, then step by the slope (see how to graph a line).
Graph the second line on the same axes.
Read the crossing point as (x, y).
Check it in both equations.

Example 1

Solve y = 2x − 2 and y = −x + 7. The lines cross at (3, 4). Check both:

y = 2x − 2: 2(3) − 2 = 4 ✓
y = −x + 7: −3 + 7 = 4 ✓
The solution is (3, 4), which means x = 3 and y = 4.

The figure below starts with exactly this system. Move the orange line with the sliders and watch the crossing, and the check, update.

Solution (3, 4)

y = 2x − 2
y = −x + 7

Line 2 slope m −1

Line 2 starts at b 7

line 1: 2(3) − 2 = 4 ✓
line 2: −1(3) + 7 = 4 ✓

The lines cross at (3, 4). Both equations give y = 4 when x = 3, so it is the solution.

Two more examples

y = x + 1 and y = −x + 5 cross at (2, 3): 2 + 1 = 3 ✓ and −2 + 5 = 3 ✓
y = 3x and y = x + 4 cross at (2, 6): 3(2) = 6 ✓ and 2 + 4 = 6 ✓

The limit of graphing

Graphing is exact when the lines cross on a grid point, and only an estimate when they do not. Try slope 1 against slope 2 in the figure and you will see a crossing between the grid lines. For those systems, use algebra: see solving a system by substitution.

Watch it explained in under a minute

The same idea, narrated step by step. Square version for sharing: download the 1:1 video.

The five mistakes to avoid

Giving only x. The solution is a point: write (3, 4), not just 3.
Trusting the picture without checking. Put the point into both equations.
Plotting one line carelessly. A slip in the slope or b moves the crossing, so recheck the dots.
Swapping the order in the point. It is always (x, y): across first, then up.
Forgetting that parallel lines never cross. Then there is no solution at all.

Try it yourself

Graph both lines, then check. Tap a question to see the answer.

1. y = x + 2 and y = −x + 6

They cross at (2, 4). Check: 2 + 2 = 4 ✓ and −2 + 6 = 4 ✓.

2. y = 2x and y = x + 3

They cross at (3, 6). Check: 2(3) = 6 ✓ and 3 + 3 = 6 ✓.

3. y = −2x + 4 and y = x − 5

They cross at (3, −2). Check: −2(3) + 4 = −2 ✓ and 3 − 5 = −2 ✓.

Where this fits in

A solution to a system is a crossing point, and this is what makes systems a graph question. Next in this course: solve a system by substitution, the exact way when the crossing is not on a grid point.