Equations with parenthesis
| Site: | Saylor Academy |
| Course: | GKT101: General Knowledge for Teachers – Math |
| Book: | Equations with parenthesis |
| Printed by: | Guest user |
| Date: | Tuesday, May 13, 2025, 11:28 PM |
Description
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
Source: Khan Academy, https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq#alg-equations-with-parentheses
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
Equations with parentheses - Questions
1. Solve for \(c\).
\(4(3+c)+c=c+4\)
2. Solve for \(p\).
\(10p−3=2(12+4p)−7\)
3. Solve for \(t\).
\(−t=9(t−10)\)
4. Solve for \(g\).
\( 6(−2g−1)=−(13g+2)\)
Answers
1.
We need to manipulate the equation to get \(c\) by itself.
| \(4(3+c)+c=c+4\) | |
| \( 12+4 c+c =c+4 \) | Distribute. |
| \( 5 c+12 =c+4 \) | Combine like terms. |
| \( 5 c+12-c =c+4-c \) | Subtract \(c\) from each side. |
| \( 4 c+12 =4 \) | Combine like terms. |
| \( 4 c+12-12 =4-12 \) | Subtract \(12\) from each side. |
| \( 4 c =-8 \) | Combine like terms. |
| \( \frac{4 c}{4} =\frac{-8}{4} \) | Divide each side by \(4\). |
| \( c =-2 \) | Simplify. |
The answer: \(c = -2\)
Let's check our work!
\(\begin{gathered}
4(3+c)+c=c+4 \\
4(3+(-2))+(-2) \stackrel{?}{=}-2+4 \\
4(1)-2 \stackrel{?}{=} 2 \\
4-2 \stackrel{?}{=} 2 \\
2=2 \quad \text { Yes! }
\end{gathered}\)
2. \(p = 10\)
We need to manipulate the equation to get \(p\) by itself.
| \(10p−3=2(12+4p)−7\) | |
| \( 10 p-3 =24+8 p-7 \) | Distribute. |
| \( 10 p-3 =17+8 p \) | Combine like terms. |
| \( 10 p-3-8 p =17+8 p-8 p\) | Subtract \(8p\) from each side. |
| \( 2 p-3 =17 \) | Combine like terms. |
| \( 2 p-3+3 =17+3 \) | Add \(3\) to each side. |
| \( 2 p =20 \) | Combine like terms. |
| \( \frac{2 p}{2} =\frac{20}{2} \) | Divide each side by \(2\). |
| \( p =10 \) | Simplify. |
The answer: \(p = 10\)
Let's check our work!
\(\begin{aligned}
10 p-3 &=2(12+4 p)-7 \\
10(10)-3 & \stackrel{?}{=} 2(12+4(10))-7 \\
100-3 & \stackrel{?}{=} 2(12+40)-7 \\
97 & \stackrel{?}{=} 2(52)-7 \\
97 & \stackrel{?}{=} 104-7 \\
97 &=97 \quad \text { Yes! }
\end{aligned}\)
3. \(t = 9\)
We need to manipulate the equation to get \(t\) by itself.
| \(−t=9(t−10)\) | |
| \( -t =9 t-90 \) | Distribute. |
| \( -t-9 t =9 t-90-9t \) | Subtract \(9t\) from each side. |
| \( -10 t =-90 \) | Combine like terms. |
| \( \frac{-10 t}{-10} =\frac{-90}{-10} \) | Divide each side by \(-10\). |
| \( t =9 \) | Simplify. |
The answer is: \(t = 9\)
Let's check our work!
\(\begin{aligned}
&-t=9(t-10) \\
&-9 \stackrel{?}{=} 9(9-10) \\
&-9 \stackrel{?}{=} 9(-1) \\
&-9=-9 \quad \text { Yes! }
\end{aligned}\)
4. \(g = 4\)
We need to manipulate the equation to get \(g\) by itself.
| \(6(−2g−1)=−(13g+2)\) | |
| \( -12 g-6 =-13 g-2 \) | Distribute. |
| \( -12 g-6+12 g =-13 g+12 g-2 \) | Add \(12g\) to each side. |
| \( -6+2 =-g-2+2 \) | Add \(2\) to each side. |
| \( -4 =-g \) | Divide by \(-1\). |
| \( 4 =g \) | Combine like terms. |
The answer: \(g = 4\)
Let's check our work!
\(\begin{aligned}
6(-2 g-1) &=-(13 g+2) \\
6(-2(4)-1) & \stackrel{?}{=}-(13(4)+2) \\
6(-8-1) & \stackrel{?}{=}-(52+2) \\
6(-9) & \stackrel{?}{=}-(54) \\
-54 &=-54 \quad \text { Yes! }
\end{aligned}\)
Equations with parentheses: decimals & fractions - Questions
1. Solve for \(t\).
Give an exact answer.
\(3 t-18=4\left(-3-\frac{3}{4} t\right)\)
2.Solve for \(b\).
Give an exact answer.
\(0.75(8b+4)−1=4b+14\)
3. Solve for \(n\).
Give an exact answer.
\(4 n+2=6\left(\frac{1}{3} n-\frac{2}{3}\right)\)
4. Solve for \(g\).
Give an exact answer.
\(12 g=12\left(\frac{2}{3} g-1\right)+11\)
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!
\(\begin{aligned}
12 g &=12\left(\frac{2}{3} g-1\right)+11 \\
12\left(-\frac{1}{4}\right) & \stackrel{?}{=} 12\left(\frac{2}{3}\left(-\frac{1}{4}\right)-1\right)+11 \\
-3 & \stackrel{?}{=} 12\left(-\frac{2}{12}-1\right)+11 \\
-3 & \stackrel{?}{=} 12\left(-\frac{2}{12}-\frac{12}{12}\right)+11 \\
-3 & \stackrel{?}{=} 12\left(-\frac{14}{12}\right)+11 \\
-3 & \stackrel{?}{=}-14+11 \\
-3 &=-3 \text { Yes!}
\end{aligned}\)