Equations with parenthesis
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!
\(\begin{aligned}
12 g &=12\left(\frac{2}{3} g-1\right)+11 \\
12\left(-\frac{1}{4}\right) & \stackrel{?}{=} 12\left(\frac{2}{3}\left(-\frac{1}{4}\right)-1\right)+11 \\
-3 & \stackrel{?}{=} 12\left(-\frac{2}{12}-1\right)+11 \\
-3 & \stackrel{?}{=} 12\left(-\frac{2}{12}-\frac{12}{12}\right)+11 \\
-3 & \stackrel{?}{=} 12\left(-\frac{14}{12}\right)+11 \\
-3 & \stackrel{?}{=}-14+11 \\
-3 &=-3 \text { Yes!}
\end{aligned}\)