Equations with variables on both sides

This lecture series shows how you can apply the principle of doing the same thing to both sides of the equation to equations with variables on both sides. Watch the videos and complete the interactive exercise sets.

Equations with variables on both sides: decimals & fractions - Questions

Answers

1. $k = 3$

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

$4.5+1.5k=18−3k$
$4.5+1.5 k+3 k =18-3 k+3 k$	Add $3k$ to each side.
$4.5 k+4.5 =18$	Combine like terms.
$4.5 k+4.5-4.5 =18-4.5$	Subtract $4.5$ from each side.
$4.5 k =13.5$	Combine like terms.
$\frac{4.5 k}{4.5} =\frac{13.5}{4.5}$	Divide each side by $4.5$ .
$k =3$	Simplify.

The answer: $k = 3$

Let's check our work!

$\begin{aligned} 4.5+1.5 k &=18-3 k \\ 4.5+1.5(3) & \stackrel{?}{=} 18-3(3) \\ 4.5+4.5 & \stackrel{?}{=} 18-9 \\ 9 &=9 \quad \text { Yes! } \end{aligned}$

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

$2-2 s=\frac{3}{4} s+13$
$2-2 s-\frac{3}{4} s =\frac{3}{4} s+13-\frac{3}{4} s$	Subtract $\frac{3}{4}$ from each side.
$-\frac{11}{4} s+2 =13$	Combine like terms.
$-\frac{11}{4} s+2-2 =13-2$	Subtract $2$ from each side.
$-\frac{11}{4} s =11$	Combine like terms.
$s \cdot\left(-\frac{4}{11}\right) =11 \cdot\left(-\frac{4}{11}\right)$	Multiply each side by $-\frac{4}{11}$ .
$s =-\frac{44}{11}$
$s =-4$	Simplify.

The answer: $s = -4$

Let's check our work!

$\begin{aligned} 2-2 s &=\frac{3}{4} s+13 \\ 2-2(-4) & \stackrel{?}{=} \frac{3}{4}(-4)+13 \\ 2+8 & \stackrel{?}{=}-\frac{12}{4}+13 \\ 10 & \stackrel{?}{=}-3+13 \\ 10 &=10 \quad \text { Yes! } \end{aligned}$

3. $g = 0.5$

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

$9+3.5g=11−0.5g$
$9+3.5 g+0.5 g =11-0.5 g+0.5 g$	Add $0.5g$ to each side.
$9+4 g =11$	Combine like terms.
$4 g+9-9 =11-9$	Subtract $9$ from each side.
$4 g =2$	Combine like terms.
$\frac{4 g}{4} =\frac{2}{4}$	Divide each side by $4$ .
$g =0.5$	Simplify.

The answer: $g = 0.5$

Let's check our work!

$\begin{aligned} 9+3.5 g &=11-0.5 g \\ 9+3.5(0.5) & \stackrel{?}{=} 11-0.5(0.5) \\ 9+1.75 & \stackrel{?}{=} 11-0.25 \\ 10.75 &=10.75 \quad \text { Yes! } \end{aligned}$

4. $p = 3$

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

$16-3 p=\frac{2}{3} p+5$
$16-3 p-\frac{2}{3} p =\frac{2}{3} p+5-\frac{2}{3} p$	Subtract $\frac{2}{3}p$ from each side.
$-\frac{11}{3} p+16 =5$	Combine like terms.
$-\frac{11}{3} p+16-16 =5-16$	Subtract $16$ from each side.
$-\frac{11}{3} p =-11$	Combine like terms.
$-\frac{11}{3} p \cdot\left(-\frac{3}{11}\right) =-11 \cdot\left(-\frac{3}{11}\right)$	Multiply each side by $-\frac{3}{11}$
$p =\frac{33}{11}$
$p =3$	Simplify.