Solving Systems by Elimination

Elimination combines equivalent equations so one variable cancels, leaving a one-variable equation.

A system’s solution satisfies every relation at once. Two lines cross at one highlighted point. Projection guides connect the intersection to both axes, showing that the same ordered pair makes both equations true.

Elimination adds or subtracts equations so opposite coefficients cancel one variable.

Why adding equations works

If two equalities are true, their left sides and right sides can be added. For a system, combining equations creates another equation satisfied by every common solution.

Opposite terms such as $3y$ and $-3y$ sum to zero, revealing the remaining variable.

A reliable method

Write equations with like terms aligned.
Multiply one or both complete equations if needed to create opposite coefficients.
Add the equations.
Solve for the remaining variable.
Substitute back to find the other variable.
Check both original equations.

Multiplying an equation means multiplying every term on both sides.

Immediate cancellation

The opposite $y$-coefficients made no scaling necessary.

Context example

Suppose $2$ adult tickets and $3$ student tickets cost $55$, while $3$ adult and $2$ student tickets cost $65$. Equations $2a+3s=55$ and $3a+2s=65$ can be scaled to eliminate either price. Solving gives $a=17$ and $s=7$ dollars.

Common mistakes

Multiplying only one term. Scale the whole equation.

Adding coefficients but not constants. Combine every aligned column and both right sides.

Forgetting back-substitution. One variable is not a full solution.

Interpreting $0=0$ as $(0,0)$. It means infinitely many solutions.

Quick self-check

Are like terms aligned?
Which coefficients can become opposites most efficiently?
Did scaling affect every term?
Have I found and checked both coordinates?