Systems of Equations & Inequalities

Systems by Substitution

Solving a system of linear equations using substitution.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Collapsing Two Equations into One

y = 2x + 3 and 3x + 2y = 16 describe two different lines. Before reading on, think about this: since both equations involve y, could you use one equation's expression for y to eliminate y from the other equation entirely — leaving just one equation with one unknown?

That's exactly the idea behind substitution: if y genuinely equals 2x + 3, then everywhere y appears in the second equation, it can be replaced by that expression instead — because they're the same value. Doing this collapses two equations with two unknowns into a single equation with only x left, solvable with tools you already have.

Worked Example — Substituting an Already-Isolated Expression

Solve y = 2x + 3 and 3x + 2y = 16. Substitute 2x + 3 for y in the second equation: 3x + 2(2x + 3) = 16. Distribute and simplify: 3x + 4x + 6 = 16, so 7x = 10, giving x = 10/7. Substitute back: y = 2(10/7) + 3 = 20/7 + 21/7 = 41/7. The solution is (10/7, 41/7).

Worked Example — Solving for a Variable Before Substituting

Solve x + y = 9 and 2x − y = 3. Neither equation has y (or x) already isolated cleanly for both — solve the first for y: y = 9 − x. Substitute into the second: 2x − (9 − x) = 3, which becomes 2x − 9 + x = 3, so 3x = 12, giving x = 4. Then y = 9 − 4 = 5. The solution is (4, 5). Check in both originals: 4 + 5 = 9 ✓, and 2(4) − 5 = 3 ✓.

Worked Example — Recognizing No Solution via Substitution

Solve y = 3x + 2 and y = 3x − 5. Substitute: 3x + 2 = 3x − 5. Subtracting 3x from both sides gives 2 = −5, a false statement. This system has no solution — the two lines share the same slope, 3, but different y-intercepts, so they're parallel and never meet.

Tip

Substitution works best when at least one equation is already solved for a variable, or can be solved for one with a coefficient of 1 — avoiding an extra fraction-heavy step before the substitution itself.

Common Mistakes

Substituting an expression into the same equation it came from, instead of into the other equation.
Solve one equation for a variable, then substitute that expression into the other equation — substituting back into the original equation just restates a true but useless statement.
Stopping after finding one variable's value and reporting it alone as the full solution.
A system's solution is a full point, (x, y) — after finding one variable, always substitute it back into an original equation to find the other.

Key Takeaways

Substitution replaces a variable with an equivalent expression from the other equation, reducing a system to one equation with one unknown.
A false statement after substitution means the system has no solution — the lines are parallel.
Substitution works most smoothly when a variable is already isolated, or easily can be.

Summary

Substitution solves a system by collapsing it to a single-variable equation. The next lesson introduces a second algebraic method — elimination — which shines exactly where substitution gets messy.

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