Lesson 1: Simplifying and Solving One-Step & Multi-Step Equations

Property

Addition and Subtraction Properties of Equality:
If the same quantity is added to or subtracted from both sides of an equation, the solution is unchanged.
In symbols, If $a = b$, then $a + c = b + c$ and $a - c = b - c$.
Multiplication and Division Properties of Equality:
If both sides of an equation are multiplied or divided by the same nonzero quantity, the solution is unchanged.
In symbols, If $a = b$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$, $c \neq 0$.

Examples

• If you have the equation $x - 5 = 10$, you can add 5 to both sides to get $x - 5 + 5 = 10 + 5$, which simplifies to $x = 15$.
• For the equation $4y = 28$, you can divide both sides by 4 to get $\frac{4y}{4} = \frac{28}{4}$, which simplifies to $y = 7$.
• If $z + 9 = 12$, you can subtract 9 from both sides to get $z + 9 - 9 = 12 - 9$, which simplifies to $z = 3$.

Explanation

To keep an equation balanced, you must perform the exact same operation on both sides. Whatever you add, subtract, multiply, or divide on one side, you must do to the other side too.

Property

We solve an equation by undoing in reverse order the operations performed on the variable. We can think of any string of terms as a sum, where the $+$ and $-$ symbols tell us the sign of the term that follows. To solve, first add or subtract terms to isolate the variable term, then multiply or divide to find the variable's value.

Examples

• To solve $12 - 4x = -8$, first subtract 12 from both sides to get $-4x = -20$. Then divide by $-4$ to find $x = 5$.
• To solve $-3y + 7 = 22$, first subtract 7 from both sides to get $-3y = 15$. Then divide by $-3$ to get $y = -5$.
• To solve $\frac{x-5}{3} = -2$, first multiply both sides by 3 to get $x-5 = -6$. Then add 5 to both sides to find $x = -1$.

Explanation

Solving an equation is like being a detective to find the variable's hidden value. You reverse the equation's steps, undoing each operation one by one until the variable is alone on one side of the equals sign.