Systems of Equations

When two (or more) linear equations share variables, you solve simultaneously via substitution or elimination. Systems abound in Riddle Solver and Variable Vault puzzles, where you infer multiple unknowns from multiple clues.

• Substitution: solve one equation for a variable, plug into the other.
• Elimination: add or subtract equations to cancel a variable, then solve.

Example: 2x + y = 5 and x − y = 1

Substitution: from the second, y = x − 1. Plug into the first: 2x + (x − 1) = 5 ⇒ 3x = 6 ⇒ x = 2

Then y = 2 − 1 = 1

Hints and tips

• Label your equations: Write (1) and (2) next to each system – never lose track of which is which.
• Choose the cleanest method: Pick substitution if one equation has a variable already isolated; pick elimination if coefficients line up.
• Scale before adding/subtracting: Multiply entire equations by constants so one variable cancels neatly.
• Check both solutions: Once you find (x, y), substitute into both original equations to confirm consistency.
• Write the answer as an ordered pair: Present (x, y) = (3, −1) clearly to avoid confusion in multi-variable puzzles.