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.


The answer: \( g = -\frac{1}{4} \)


Let's check our work!

\(\begin{aligned}

12 g &=12\left(\frac{2}{3} g-1\right)+11 \\

12\left(-\frac{1}{4}\right) & \stackrel{?}{=} 12\left(\frac{2}{3}\left(-\frac{1}{4}\right)-1\right)+11 \\

-3 & \stackrel{?}{=} 12\left(-\frac{2}{12}-1\right)+11 \\

-3 & \stackrel{?}{=} 12\left(-\frac{2}{12}-\frac{12}{12}\right)+11 \\

-3 & \stackrel{?}{=} 12\left(-\frac{14}{12}\right)+11 \\

-3 & \stackrel{?}{=}-14+11 \\

-3 &=-3 \text { Yes!}

\end{aligned}\)