EQN-03-04

Systems of Two Equations

Two unknowns, two equations — solve by substitution or elimination, whichever is faster.

~5 min read•Difficulty 3/5

When you need a system

Two unknowns need two independent equations. The solution is the single pair (x, y) that satisfies both.

Method 1 — Substitution

Solve one equation for a variable, then substitute into the other.

y = x + 1

2 x + y = 7 \to 2 x + (x + 1) = 7 \to 3 x = 6 \to x = 2, y = 3

Method 2 — Elimination

Add or subtract the equations to cancel a variable. Scale first if needed so one variable matches.

3 x + 2 y = 16

3 x - y = 4 subtract \to 3 y = 12 \to y = 4, then x = 8/3...

(Pick the variable that cancels most cleanly.)

A quick shortcut for the GAT-E

If the question asks for $x + y$ or $x - y$ rather than each variable, adding or subtracting the two equations often gives it directly — no need to solve for x and y separately.

Special cases

Same line (identical after scaling $) \to$ infinitely many solutions.
Parallel lines (same left side, different right side $) \to$ no solution.

Worked examples

Example 1 (difficulty 2/5)

Solve

: x + y = 10

and

x - y = 4.

What is x ?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

2 x + 3 y = 12

and

2 x + y = 8,

what is y ?

◎ Predict the answer first:

Tips

Choose elimination when a variable already has matching coefficients; substitution when one variable is already isolated.
If asked for $x + y$ or $x - y,$ try adding/subtracting the equations directly — often a one-step answer.
Always check your pair in BOTH original equations.

Common mistakes

Solving for one variable and forgetting to find the other

Why: The first value feels like "the answer".

Fix: Read what the question asks; substitute back to get the second value if needed.

Adding equations when subtracting would cancel the variable

Why: Adding is the default reflex.

Fix: Match the signs: same sign

\to

subtract to cancel; opposite signs

\to

add.

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