Equations with parenthesis

Finally, you will look at the most general linear equations with one variable: equations involving parentheses. Here, you have to simplify each side by opening parentheses before attempting to solve by doing the same thing to both sides. Watch this lecture series and complete the interactive exercises.

Equations with parentheses: decimals & fractions - Questions

Answers

1. $t = 1$

We need to manipulate the equation to get $t$ by itself.

$3 t-18=4\left(-3-\frac{3}{4} t\right)$
$3 t-18 =-12-3 t$	Distribute.
$3 t-18+3 t =-12-3 t+3 t$	Add $3t$ to each side.
$6 t-18 =-12 = 18$	Combine like terms.
$6 t-18+18 =-12+18$	Add $18$ to each side.
$6 t =6$	Combine like terms.
$\frac{6 t}{6} =\frac{6}{6}$	Divide each side by $6$ .
$t =1$	Simplify.

The answer: $t = 1$

Let's check our work!

$\begin{aligned} 3 t-18 &=4\left(-3-\frac{3}{4} t\right) \\ 3(1)-18 & \stackrel{?}{=} 4\left(-3-\frac{3}{4}(1)\right) \\ 3-18 & \stackrel{?}{=} 4\left(-3-\frac{3}{4}\right) \\ -15 & \stackrel{?}{=} 4\left(-\frac{12}{4}-\frac{3}{4}\right) \\ -15 & \stackrel{?}{=} 4\left(-\frac{15}{4}\right) \\ -15 & \stackrel{?}{=}-\frac{60}{4} \\ -15 &=-15 \text { Yes!} \end{aligned}$

2. $b = 6$

We need to manipulate the equation to get $b$ by itself.

$0.75(8b+4)−1=4b+14$
$6 b+3-1 =4 b+14$	Distribute.
$6 b+2 =4 b+14$	Combine like terms.
$6 b+2-4 b =4 b+14 - 4b$	Subtract $4b$ from each side.
$2 b + 2 = 14$	Combine like terms.
$2 b+2-2 =14-2$	Subtract $2$ from each side.
$2 b =12$	Combine like terms.
$\frac{2 b}{2} =\frac{12}{2}$	Divide each side by $2$ .
$b =6$	Simplify.

The answer: $b = 6$

Let's check our work!

$\begin{gathered} 0.75(8 b+4)-1=4 b+14 \\ 0.75(8(6)+4)-1 \stackrel{?}{=} 4(6)+14 \\ 0.75(48+4)-1 \stackrel{?}{=} 24+14 \\ 0.75(52)-1 \stackrel{?}{=} 38 \\ 39-1 \stackrel{?}{=} 38 \\ 38=38 \quad \text { Yes! } \end{gathered}$

3. $n = -3$

We need to manipulate the equation to get $n$ by itself.

$4 n+2=6\left(\frac{1}{3} n-\frac{2}{3}\right)$
$4 n+2 =2 n-4$	Distribute.
$4 n+2-2 n =2 n-4-2 n$	Subtract $2n$ from each side.
$2 n+2 =-4$	Combine like terms.
$2 n+2-2 =-4-2$	Subtract $2$ from each side.
$2 n =-6$	Combine like terms.
$\frac{2 n}{2} =\frac{-6}{2}$	Divide each side by $2$ .
$n =-3$	Simplify.

The answer: $n = -3$

Let's check our work!

$\begin{aligned} 4 n+2 &=6\left(\frac{1}{3} n-\frac{2}{3}\right) \\ 4(-3)+2 & \stackrel{?}{=} 6\left(\frac{1}{3}(-3)-\frac{2}{3}\right) \\ -12+2 & \stackrel{?}{=} 6\left(-1-\frac{2}{3}\right) \\ -10 & \stackrel{?}{=} 6\left(-\frac{5}{3}\right) \\ -10 & \stackrel{?}{=}-\frac{30}{3} \\ -10 &=-10 \text { Yes!} \end{aligned}$

4. $g = -\frac{1}{4}$

We need to manipulate the equation to get $g$ by itself.

$12 g=12\left(\frac{2}{3} g-1\right)+11$
$12 g =8 g-12+11$	Distribute.
$12 g =8 g-1$	Combine like terms.
$12 g-8 g =8 g-1-8 g$	Subtract $8g$ from each side.
$4 g =-1$	Combine like terms.
$\frac{4 g}{4} =\frac{-1}{4}$	Divide each side by $4$ .
$g =-\frac{1}{4}$	Simplify.