Systems of equations: learn and practice

Who this lesson is for

This systems lesson is for grade 8. Students find one pair (x, y) that makes two linear equations true at the same time. A single linear equation is the grades 7–8 lesson on its own page. The catalog continues systems into grade 9, and the published hubs stop at grade 8, so only that hub links here.

One pair has to fit both lines

One equation in two unknowns has many solutions. A second equation cuts that list down to one pair, when the lines cross once. The pair you report has to be substituted into both equations, not only the one you used last. If it fails either equation, it is not the solution of the system.

Remove one unknown, then come back

Line the equations up and eliminate one variable. You can add or subtract the equations when a variable has matching or opposite coefficients, or you can solve one equation for a variable and substitute. After you know one value, substitute it back to find the other. The catalog hint is the same: eliminate one variable, then back-substitute.

Worked example 1: x + y = 10 and x − y = 2

Solve x + y = 10 and x − y = 2. Add the equations. The y terms cancel: 2x = 12, so x = 6. Substitute into the first equation: 6 + y = 10, so y = 4. Check the second equation: 6 − 4 = 2, which matches. The solution is x = 6, y = 4. Using only the first equation, x = 6 and y = 4 is one of many pairs that sum to 10. The second equation is what picks this pair out.

Worked example 2: 2x + y = 7 and x + y = 4

Solve 2x + y = 7 and x + y = 4. Subtract the second from the first. The y terms cancel: x = 3. Substitute into x + y = 4: 3 + y = 4, so y = 1. Check the first equation: 2 × 3 + 1 = 7. Both equations hold. The solution is x = 3, y = 1. Subtracting in the other order gives −x = −3, which is the same x.

Worked example 3: x + 2y = 8 and x + y = 5

Solve x + 2y = 8 and x + y = 5. Subtract the second from the first: y = 3. Substitute into x + y = 5: x + 3 = 5, so x = 2. Check the first equation: 2 + 2 × 3 = 8. The solution is x = 2, y = 3. Stopping after finding y and reporting only y leaves the system unfinished. Both unknowns belong in the answer.

Checks before you leave the problem

Substitute the pair into both equations. Each side should match. If you eliminated a variable, the back-substitution is part of the answer, not an optional check. Then practice systems on the board.

Common questions about systems of equations

Can two equations have more than one solution?

A pair of lines can be parallel and never meet, or they can be the same line. The exercises on this page are built so the lines meet once, and the answer is one pair.

Do I have to write both x and y?

Yes. A system asks for the pair. x = 6 alone does not say which point on that vertical line was intended. The catalog answer looks like x = 6, y = 4.

How is this different from one linear equation?

One equation such as 2x + 3 = 11 has a single unknown. A system has two unknowns and two equations that have to be true together. That single-equation skill stays on the linear equations page.