# 5.3 Further techniques in equation solving  (Page 2/2)

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Solve $3\left(m-6\right)-2m=-4+1$ for $m.$

$\begin{array}{lll}\hfill 3\left(m-6\right)-2m& =\hfill & -4+1\hfill \\ \hfill 3m-18-2m& =\hfill & -3\hfill \\ \hfill m-18& =\hfill & -3\hfill \\ \hfill m& =\hfill & 15\hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 3\left(15-6\right)-2\left(15\right)& =\hfill & -4+1\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 3\left(9\right)-30& =\hfill & -3\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 27-30& =\hfill & -3\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -3& =\hfill & -3\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

## Practice set b

Solve and check each equation.

$16x-3-15x=8$ for $x.$

$x=11$

$4\left(y-5\right)-3y=-1$ for $y.$

$y=19$

$-2\left({a}^{2}+3a-1\right)+2{a}^{2}+7a=0$ for $a.$

$a=-2$

$5m\left(m-2a-1\right)-5{m}^{2}+2a\left(5m+3\right)=10$ for $a.$

$a=\frac{10+5m}{6}$

Often the variable we wish to solve for will appear on both sides of the equal sign. We can isolate the variable on either the left or right side of the equation by using the techniques of Sections [link] and [link] .

## Sample set c

Solve $6x-4=2x+8$ for $x.$

$\begin{array}{llll}\hfill 6x-4& =\hfill & 2x+8\hfill & \text{To}\text{\hspace{0.17em}}\text{isolate}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{left}\text{\hspace{0.17em}}\text{side,}\text{\hspace{0.17em}}\text{subtract}\text{\hspace{0.17em}}2m\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill 6x-4-2x& =\hfill & 2x+8-2x\hfill & \hfill \\ \hfill 4x-4& =\hfill & 8\hfill & \text{Add}\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill 4x-4+4& =\hfill & 8+4\hfill & \hfill \\ \hfill 4x& =& 12\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill \frac{4x}{4}& =\hfill & \frac{12}{4}\hfill & \\ \hfill x& =\hfill & 3\hfill & \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 6\left(3\right)-4& =\hfill & 2\left(3\right)+8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 18-4& =\hfill & 6+8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 14& =\hfill & 14\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Solve $6\left(1-3x\right)+1=2x-\left[3\left(x-7\right)-20\right]$ for $x.$

$\begin{array}{llll}\hfill 6-18x+1& =\hfill & 2x-\left[3x-21-20\right]\hfill & \hfill \\ \hfill -18x+7& =\hfill & 2x-\left[3x-41\right]\hfill & \\ \hfill -18x+7& =\hfill & 2x-3x+41\hfill & \\ \hfill -18x+7& =\hfill & -x+41\hfill & \text{To}\text{\hspace{0.17em}}\text{isolate}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{right}\text{\hspace{0.17em}}\text{side,}\text{\hspace{0.17em}}\text{add}\text{\hspace{0.17em}}18x\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill -18x+7+18x& =\hfill & -x+41+18x\hfill & \hfill \\ \hfill 7& =\hfill & 17x+41\hfill & \text{Subtract}\text{\hspace{0.17em}}41\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill 7-41& =\hfill & 17x+41-41\hfill & \hfill \\ \hfill -34& =\hfill & 17x\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}17.\hfill \\ \hfill \frac{-34}{17}& =\hfill & \frac{17x}{17}\hfill & \\ \hfill -2& =\hfill & x\hfill & \text{Since}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{equation}\text{\hspace{0.17em}}-2=x\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{equivalent}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{equation}\hfill \\ \hfill & \hfill & \hfill & x=-2,\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{write}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{answer}\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x=-2.\hfill \\ \hfill x& =\hfill & -2\hfill & \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 6\left(1-3\left(-2\right)\right)+1& =\hfill & 2\left(-2\right)-\left[3\left(-2-7\right)-20\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 6\left(1+6\right)+1& =\hfill & -4-\left[3\left(-9\right)-20\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 6\left(7\right)+1& =\hfill & -4-\left[-27-20\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 42+1& =\hfill & -4-\left[-47\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 43& =\hfill & -4+47\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 43& =\hfill & 43\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

## Practice set c

Solve $8a+5=3a-5$ for $a.$

$a=-2$

Solve $9y+3\left(y+6\right)=15y+21$ for $y.$

$y=-1$

Solve $3k+2\left[4\left(k-1\right)+3\right]=63-2k$ for $k.$

$k=5$

As we noted in Section [link] , some equations are identities and some are contradictions. As the problems of Sample Set D will suggest,

## Recognizing an identity

1. If, when solving an equation, all the variables are eliminated and a true statement results, the equation is an identity.

1. If, when solving an equation, all the variables are eliminated and a false statement results, the equation is a contradiction.

## Sample set d

Solve $9x+3\left(4-3x\right)=12$ for $x.$

$\begin{array}{lll}\hfill 9x+12-9x& =\hfill & 12\hfill \\ \hfill 12& =\hfill & 12\hfill \end{array}$

The variable has been eliminated and the result is a true statement. The original equation is an identity.

Solve $-2\left(10-2y\right)-4y+1=-18$ for $y.$

$\begin{array}{lll}\hfill -20+4y-4y+1& =\hfill & -18\hfill \\ \hfill -19& =\hfill & -18\hfill \end{array}$

The variable has been eliminated and the result is a false statement. The original equation is a contradiction.

## Practice set d

Classify each equation as an identity or a contradiction.

$6x+3\left(1-2x\right)=3$

identity, $3=3$

$-8m+4\left(2m-7\right)=28$

contradiction, $-28=28$

$3\left(2x-4\right)-2\left(3x+1\right)+14=0$

identity, $0=0$

$-5\left(x+6\right)+8=3\left[4-\left(x+2\right)\right]-2x$

contradiction, $-22=6$

## Exercises

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction.

$3x+1=16$

$x=5$

$6y-4=20$

$4a-1=27$

$a=7$

$3x+4=40$

$2y+7=-3$

$y=-5$

$8k-7=-23$

$5x+6=-9$

$x=-3$

$7a+2=-26$

$10y-3=-23$

$y=-2$

$14x+1=-55$

$\frac{x}{9}+2=6$

$x=36$

$\frac{m}{7}-8=-11$

$\frac{y}{4}+6=12$

$y=24$

$\frac{x}{8}-2=5$

$\frac{m}{11}-15=-19$

$m=-44$

$\frac{k}{15}+20=10$

$6+\frac{k}{5}=5$

$k=-5$

$1-\frac{n}{2}=6$

$\frac{7x}{4}+6=-8$

$x=-8$

$\frac{-6m}{5}+11=-13$

$\frac{3k}{14}+25=22$

$k=-14$

$3\left(x-6\right)+5=-25$

$16\left(y-1\right)+11=-85$

$y=-5$

$6x+14=5x-12$

$23y-19=22y+1$

$y=20$

$-3m+1=3m-5$

$8k+7=2k+1$

$k=-1$

$12n+5=5n-16$

$2\left(x-7\right)=2x+5$

$-4\left(5y+3\right)+5\left(1+4y\right)=0$

$3x+7=-3-\left(x+2\right)$

$x=-3$

$4\left(4y+2\right)=3y+2\left[1-3\left(1-2y\right)\right]$

$5\left(3x-8\right)+11=2-2x+3\left(x-4\right)$

$x=\frac{19}{14}$

$12-\left(m-2\right)=2m+3m-2m+3\left(5-3m\right)$

$-4\cdot k-\left(-4-3k\right)=-3k-2k-\left(3-6k\right)+1$

$k=3$

$3\left[4-2\left(y+2\right)\right]=2y-4\left[1+2\left(1+y\right)\right]$

$-5\left[2m-\left(3m-1\right)\right]=4m-3m+2\left(5-2m\right)+1$

$m=2$

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables.

Solve $I=\frac{E}{R}$ for $R.$ Find the value of $R$ when $I=0.005$ and $E=0.0035.$

Solve $P=R-C$ for $R.$ Find the value of $R$ when $P=27$ and $C=85.$

$R=112$

Solve $z=\frac{x-\overline{x}}{s}$ for $x.$ Find the value of $x$ when $z=1.96,$ $s=2.5,$ and $\overline{x}=15.$

Solve $F=\frac{{S}_{x}^{2}}{{S}_{y}^{2}}$ for ${S}_{x}^{2}\cdot {S}_{x}^{2}$ represents a single quantity. Find the value of ${S}_{x}^{2}$ when $F=2.21$ and ${S}_{y}^{2}=3.24.$

${S}_{x}{}^{2}=F·{S}_{y}{}^{2};\text{\hspace{0.17em}}{S}_{x}{}^{2}=7.1604$

Solve $p=\frac{nRT}{V}$ for $R.$

Solve $x=4y+7$ for $y.$

$y=\frac{x-7}{4}$

Solve $y=10x+16$ for $x.$

Solve $2x+5y=12$ for $y.$

$y=\frac{-2x+12}{5}$

Solve $-9x+3y+15=0$ for $y.$

Solve $m=\frac{2n-h}{5}$ for $n.$

$n=\frac{5m+h}{2}$

Solve $t=\frac{Q+6P}{8}$ for $P.$

Solve $=\frac{\square \text{\hspace{0.17em}}+9j}{\Delta }$ for $j$ .

Solve for .

## Exercises for review

( [link] ) Simplify ${\left(x+3\right)}^{2}{\left(x-2\right)}^{3}{\left(x-2\right)}^{4}\left(x+3\right).$

${\left(x+3\right)}^{3}{\left(x-2\right)}^{7}$

( [link] ) Find the product. $\left(x-7\right)\left(x+7\right).$

( [link] ) Find the product. ${\left(2x-1\right)}^{2}.$

$4{x}^{2}-4x+1$

( [link] ) Solve the equation $y-2=-2.$

( [link] ) Solve the equation $\frac{4x}{5}=-3.$

$x=\frac{-15}{4}$

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