# 2.3E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Practice Makes Perfect

Solve Equations with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

##### Exercise $$\PageIndex{1}$$

$$9 x-3=60$$

##### Exercise $$\PageIndex{2}$$

$$12 x-8=64$$

$$x=6$$

##### Exercise $$\PageIndex{3}$$

$$14 w+5=117$$

##### Exercise $$\PageIndex{4}$$

$$15 y+7=97$$

$$y=6$$

##### Exercise $$\PageIndex{5}$$

$$2 a+8=-28$$

##### Exercise $$\PageIndex{6}$$

$$3 m+9=-15$$

$$m=-8$$

##### Exercise $$\PageIndex{7}$$

$$-62=8 n-6$$

##### Exercise $$\PageIndex{8}$$

$$-77=9 b-5$$

$$b=-8$$

##### Exercise $$\PageIndex{9}$$

$$35=-13 y+9$$

##### Exercise $$\PageIndex{10}$$

$$60=-21 x-24$$

$$x=-4$$

##### Exercise $$\PageIndex{11}$$

$$-12 p-9=9$$

##### Exercise $$\PageIndex{12}$$

$$-14 q-2=16$$

$$q=-\frac{9}{7}$$

Solve Equations with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

##### Exercise $$\PageIndex{13}$$

$$19 z=18 z-7$$

##### Exercise $$\PageIndex{14}$$

$$21 k=20 k-11$$

$$k=-11$$

##### Exercise $$\PageIndex{15}$$

$$9 x+36=15 x$$

##### Exercise $$\PageIndex{16}$$

$$8 x+27=11 x$$

$$x=9$$

##### Exercise $$\PageIndex{17}$$

$$c=-3 c-20$$

##### Exercise $$\PageIndex{18}$$

$$b=-4 b-15$$

$$b=-3$$

##### Exercise $$\PageIndex{19}$$

$$9 q=44-2 q$$

##### Exercise $$\PageIndex{20}$$

$$5 z=39-8 z$$

$$z=3$$

##### Exercise $$\PageIndex{21}$$

$$6 y+\frac{1}{2}=5 y$$

##### Exercise $$\PageIndex{22}$$

$$4 x+\frac{3}{4}=3 x$$

$$x=-\frac{3}{4}$$

##### Exercise $$\PageIndex{23}$$

$$-18 a-8=-22 a$$

##### Exercise $$\PageIndex{24}$$

$$-11 r-8=-7 r$$

$$r=-2$$

Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

##### Exercise $$\PageIndex{25}$$

$$8 x-15=7 x+3$$

##### Exercise $$\PageIndex{26}$$

$$6 x-17=5 x+2$$

$$x=19$$

##### Exercise $$\PageIndex{27}$$

$$26+13 d=14 d+11$$

##### Exercise $$\PageIndex{28}$$

$$21+18 f=19 f+14$$

$$f=7$$

##### Exercise $$\PageIndex{29}$$

$$2 p-1=4 p-33$$

##### Exercise $$\PageIndex{30}$$

$$12 q-5=9 q-20$$

$$q=-5$$

##### Exercise $$\PageIndex{31}$$

$$4 a+5=-a-40$$

##### Exercise $$\PageIndex{32}$$

$$8 c+7=-3 c-37$$

$$c=-4$$

##### Exercise $$\PageIndex{33}$$

$$5 y-30=-5 y+30$$

##### Exercise $$\PageIndex{34}$$

$$7 x-17=-8 x+13$$

$$x=2$$

##### Exercise $$\PageIndex{35}$$

$$7 s+12=5+4 s$$

##### Exercise $$\PageIndex{36}$$

$$9 p+14=6+4 p$$

$$p=-\frac{8}{5}$$

##### Exercise $$\PageIndex{37}$$

$$2 z-6=23-z$$

##### Exercise $$\PageIndex{38}$$

$$3 y-4=12-y$$

$$y=4$$

##### Exercise $$\PageIndex{39}$$

$$\frac{5}{3} c-3=\frac{2}{3} c-16$$

##### Exercise $$\PageIndex{40}$$

$$\frac{7}{4} m-7=\frac{3}{4} m-13$$

$$m=-6$$

##### Exercise $$\PageIndex{41}$$

$$8-\frac{2}{5} q=\frac{3}{5} q+6$$

##### Exercise $$\PageIndex{42}$$

$$11-\frac{1}{5} a=\frac{4}{5} a+4$$

$$a=7$$

##### Exercise $$\PageIndex{43}$$

$$\frac{4}{3} n+9=\frac{1}{3} n-9$$

##### Exercise $$\PageIndex{44}$$

$$\frac{5}{4} a+15=\frac{3}{4} a-5$$

$$a=-40$$

##### Exercise $$\PageIndex{45}$$

$$\frac{1}{4} y+7=\frac{3}{4} y-3$$

##### Exercise $$\PageIndex{46}$$

$$\frac{3}{5} p+2=\frac{4}{5} p-1$$

$$p=15$$

##### Exercise $$\PageIndex{47}$$

$$14 n+8.25=9 n+19.60$$

##### Exercise $$\PageIndex{48}$$

$$13 z+6.45=8 z+23.75$$

$$z=3.46$$

##### Exercise $$\PageIndex{49}$$

$$2.4 w-100=0.8 w+28$$

##### Exercise $$\PageIndex{50}$$

$$2.7 w-80=1.2 w+10$$

$$w=60$$

##### Exercise $$\PageIndex{51}$$

$$5.6 r+13.1=3.5 r+57.2$$

##### Exercise $$\PageIndex{52}$$

$$6.6 x-18.9=3.4 x+54.7$$

$$x=23$$

## Everyday Math

##### Exercise $$\PageIndex{53}$$

Concert tickets At a school concert the total value of tickets sold was $1506. Student tickets sold for$6 and adult tickets sold for \$9. The number of adult tickets sold was 5 less than 3 times the number of student tickets. Find the number of student tickets sold, s, by solving the equation 6s+27s−45=1506. Add exercises text here.

##### Exercise $$\PageIndex{54}$$

Making a fence Jovani has 150 feet of fencing to make a rectangular garden in his backyard. He wants the length to be 15
feet more than the width. Find the width, w, by solving the equation $$150=2 w+30+2 w$$.

30 feet

## Writing Exercises

##### Exercise $$\PageIndex{55}$$

Solve the equation $$\frac{6}{5} y-8=\frac{1}{5} y+7$$ explaining all the steps of your solution as in the examples in this section.

##### Exercise $$\PageIndex{56}$$

Solve the equation $$10 x+14=-2 x+38$$ explaining all the steps of your solution as in this section.

$$x=2$$ Justifications will vary.

##### Exercise $$\PageIndex{57}$$

When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient
of $$x$$ to be the "variable" side?

##### Exercise $$\PageIndex{58}$$

Is $$x=-2$$ a solution to the equation $$5-2 x=-4 x+1 ?$$ How do you know?