Systems of Equations: Solving by Substitution, Elimination and Graphing

Q: Should I use substitution or elimination?

Use substitution when one equation already has x or y on its own. Use elimination when the coefficients of one variable match or are easy to match.

What it means to solve two equations together, how the solution appears on a graph, the substitution and elimination methods step by step, and when a system has no solution or infinitely many.

3 min read Supports 3 lessons

One equation with two unknowns, like $x + y = 10$ , has endless solutions: $(1, 9)$ , $(4, 6)$ , $(7, 3)$ , and so on. Add a second condition, $x - y = 4$ , and only one pair satisfies both: $(7, 3)$ .

Each linear equation is a line. A point on the first line satisfies the first equation; a point on the second satisfies the second. The point on both lines, where they cross, satisfies both. Adjust the lines below:

Elimination

Add or subtract the equations so that one variable cancels out.

$\begin{aligned} x + y &= 10 \\ x - y &= 4 \end{aligned}$
The $y$ terms have opposite signs.

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Tip

If no variable cancels straight away, multiply one or both equations first so the coefficients match. To eliminate $y$ from $2x + 3y = 12$ and $x + y = 5$ , multiply the second by $3$ .

Substitution

If one equation already says what $x$ or $y$ equals, substitute that expression into the other equation.

Worked example

Solve $y = 2x - 1$ and $3x + y = 14$ .

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Replace $y$ in the second equation.
$3x + (2x - 1) = 14$

Key idea

Both methods do the same job: they turn two equations in two unknowns into one equation in one unknown. Choose whichever makes that easier.

No solution, or infinitely many

Two lines usually meet once. But parallel lines never meet, so the system has no solution; the algebra collapses to something false, like $0 = 5$ . And if the two equations describe the same line, every point on it works, giving infinitely many solutions; the algebra collapses to something always true, like $0 = 0$ .

Same slope, different intercepts: parallel lines, no solution.

Watch out

Always find both values. A solution to a system is a pair $(x, y)$ , not just one number.

Common questions

Should I use substitution or elimination?: Use substitution when one equation already has $x$ or $y$ on its own. Use elimination when the coefficients of one variable match or are easy to match.
How can a system have no solution?: If the two lines are parallel they never cross, so no point lies on both. Algebraically you end up with a false statement such as $0 = 5$ .

Practise systems of equations

The Systems of Linear Equations lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Should I use substitution or elimination?

Use substitution when one equation already has $x$ or $y$ on its own. Use elimination when the coefficients of one variable match or are easy to match.

How can a system have no solution?

If the two lines are parallel they never cross, so no point lies on both. Algebraically you end up with a false statement such as $0 = 5$ .