Lesson 2: Solving Equations of the Form px+q=r and p(x+q)=r and Problems That Lead to Those 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

A balanced hanger represents an equation where the total weight on the left side is equal to the total weight on the right side.
If the left side has weight $L$ and the right side has weight $R$, the balanced hanger represents the equation $L = R$.

Examples

Property

To solve an equation of the form $px + q = r$, use inverse operations to isolate the variable. First, undo the addition or subtraction of the constant term $q$. Then, undo the multiplication by the coefficient $p$.

Examples