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Equations with parenthesis

Site: Saylor Academy
Course: GKT101: General Knowledge for Teachers – Math
Book: Equations with parenthesis
Printed by: Guest user
Date: Friday, 4 April 2025, 11:12 AM

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.

Table of contents

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}