Lesson 8-2: Solve Equations: p(x + q) = r

Property

Both sides of an equation can be divided by the same number, and the statement will still be true.
If $a=b$, then $\frac{a}{c} = \frac{b}{c}$ (where $c \ne 0$).

Examples

To solve $5x = 20$, divide both sides by 5: $\frac{5x}{5} = \frac{20}{5}$, which simplifies to $x = 4$.
In the equation $-12 = 3n$, divide both sides by 3: $\frac{-12}{3} = \frac{3n}{3}$, which gives you $-4 = n$.

Explanation

Just like with multiplying, you have to keep things fair and balanced! If you divide one side of your equation by a number, you must do the exact same thing to the other side. This ensures your equation stays equal, helping you march steadily toward finding the value of your hidden variable.

Property

$a(b + c) = ab + ac$
$a(b - c) = ab - ac$

Examples

Show that $2(l+w)$ and $2l+2w$ are equal for $l=30, w=20$: $2(30+20)=2(50)=100$ and $2(30)+2(20)=60+40=100$.
Simplify using the property: $2(n+5) = 2 \cdot n + 2 \cdot 5 = 2n+10$.

Explanation

This property is your ticket to breaking open parentheses! The term outside gets "distributed" or multiplied by every single term inside. It's like a pizza delivery—the number outside brings a slice of multiplication to everyone waiting inside the parentheses. No one gets left out!

Property

An equation of the form $p(x+q)=r$ can be solved using two different, equally valid methods:

1. Divide First: Divide both sides by $p$ to get $x+q = \frac{r}{p}$.
2. Distribute First: Apply the distributive property to get $px+pq = r$.

Examples