Systems of Linear Equations

Substitution

Solve one equation for a variable and substitute into the other.

$y = 2x + 1$ and $3x + y = 11$

$3x + (2x + 1) = 11$ → $5x = 10$ → $x = 2$, $y = 5$.

Elimination

Add or subtract equations to eliminate one variable.

$2x + 3y = 12$ and $4x - 3y = 6$

Add: $6x = 18$ → $x = 3$. Sub back: $y = 2$.

Graphical

The solution is the intersection point. If lines are parallel (same slope, different intercept), no solution. If identical, infinitely many solutions.

• Elimination is often faster than substitution on the ACT.
• If asked for an expression like $x + y$ rather than individual values, you may be able to find it directly.