Lesson 3: Solving Systems of Equations by Elimination

Property

For any numbers $a$, $b$, and $c$,

When you add the same quantity to both sides of an equation, you still have equality.

Examples

• To solve $p - 8 = 14$, add 8 to both sides: $p - 8 + 8 = 14 + 8$, which simplifies to $p = 22$.
• To solve $q - \frac{1}{5} = \frac{3}{5}$, add $\frac{1}{5}$ to both sides: $q - \frac{1}{5} + \frac{1}{5} = \frac{3}{5} + \frac{1}{5}$, which simplifies to $q = \frac{4}{5}$.
• To solve $r - 5.5 = -10$, add 5.5 to both sides: $r - 5.5 + 5.5 = -10 + 5.5$, which simplifies to $r = -4.5$.

Explanation

This property helps you 'undo' subtraction in an equation. By adding the same number to both sides, you keep the equation balanced and get the variable all by itself.

Property

Multiplication Property of Equality: If both sides of an equation are multiplied by the same nonzero quantity, the solution is unchanged. In symbols, if $a = b$, then $ac = bc$ where $c \neq 0$.

Examples

Property

When using elimination, choose the variable that requires the least work to create opposite coefficients.
Look for variables where coefficients are already opposites, have a common factor, or where one coefficient is 1.

Examples