System of 2 equations, 2 unknowns: substitution that always works

The 3 steps

To solve: $\begin{cases} x + 2y = 8 \\ 3x - y = 3 \end{cases}$

Step 1 — Isolate ONE unknown in ONE equation (pick the simplest).

Equation 1: $x = 8 - 2y$.

Step 2 — Substitute in the OTHER equation.

Equation 2: $3(8 - 2y) - y = 3$ $24 - 6y - y = 3$ $-7y = -21$ $y = 3$.

Step 3 — Back-solve for the first unknown.

$x = 8 - 2 \times 3 = 2$.

Solution: $(x, y) = (2, 3)$.

Check (essential)

• $2 + 2 \times 3 = 8$ ✓
• $3 \times 2 - 3 = 3$ ✓

How to pick “the simplest” to isolate?

Look for a coefficient equal to 1 or −1. Avoids fractions.

• In $x + 2y = 8$, $x$ has coefficient 1 → perfect, isolate $x$.
• In $3x + 5y = 12$, no coefficient is 1 → prefer the elimination method.

Special cases

• If you reach $0 = 5$ or similar: no solution (parallel lines).
• If you reach $0 = 0$: infinitely many (same line twice).

?À toi de jouer

Solve x − y = 1 and 2x + y = 8. What is y?

A.2B.3C.1D.4

Valider