5.3 Further techniques in equation solving

Solve $4x-7=9$ for $x.$

$\begin{array}{llll}\hfill 4x-7& =\hfill & 9\hfill & \text{First,}\text{undo}\text{the}\text{association}\text{between}x\text{and}7.\hfill \\ \hfill & \hfill & \hfill & \text{The}7\text{is}\text{associated}\text{with}x\text{by}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{the}\text{association}\text{by}\text{adding}7\text{to}both\text{sides}\text{.}\hfill \\ \hfill 4x-7+7& =\hfill & 9+7\hfill & \hfill \\ \hfill 4x& =\hfill & 16\hfill & \text{Now,}\text{undo}\text{the}\text{association}\text{between}x\text{and}4.\hfill \\ \hfill & \hfill & \hfill & \text{The}4\text{is}\text{associated}\text{with}x\text{by}\text{multiplication}.\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{the}\text{association}\text{by}\text{dividing}both\text{sides}\text{by}4.\hfill \\ \hfill \frac{4x}{4}& =& \frac{16}{4}& \\ \hfill 16-7& =\hfill & 9\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill x& =\hfill & 4\hfill & \hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 4(4)-7& =\hfill & 9\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill & \hfill 9& =\hfill & 9\hfill & \text{Yes,}\text{this}\text{is}\text{correct}\text{.}\hfill \end{array}$

Solve $\frac{3y}{4}-5=-11.$

$\begin{array}{llll}\hfill \frac{3y}{4}-5& =\hfill & -11\hfill & -5\text{is}\text{associated}\text{with}y\text{by}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{the}\text{association}\text{by}\text{adding}5\text{to}both\text{sides}\text{.}\hfill \\ \hfill \frac{3y}{4}-5+5& =\hfill & -11+5\hfill & \hfill \\ \hfill \frac{3y}{4}& =\hfill & -6\hfill & 4\text{is}\text{associated}\text{with}y\text{by}\text{division}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{the}\text{association}\text{by}\text{multiplying}both\text{sides}\text{by}4.\hfill \\ \hfill 4\cdot \frac{3y}{4}& =\hfill & 4(-6)\hfill & \hfill \\ \hfill 4\cdot \frac{3y}{4}& =\hfill & 4(-6)\hfill & \hfill \\ \hfill 3y& =\hfill & -24\hfill & 3\text{is}\text{associated}\text{with}y\text{by}\text{multiplication}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{the}\text{association}\text{by}\text{dividing}both\text{sides}\text{by}3.\hfill \\ \hfill \frac{3y}{3}& =& \frac{-24}{3}& \hfill \\ \hfill \frac{3y}{3}& =\hfill & -8\hfill & \hfill \\ \hfill y& =\hfill & -8\hfill & \hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill \frac{3(-8)}{4}-5& =\hfill & -11\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill & \hfill \frac{-24}{4}-5& =\hfill & -11\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill & \hfill -6-5& =\hfill & -11\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill & \hfill -11& =\hfill & -11\hfill & \text{Yes,}\text{this}\text{is}\text{correct}\text{.}\hfill \end{array}$

Solve $\frac{8a}{3b}+2m=6m-5$ for $a.$

$\begin{array}{llll}\hfill \frac{8a}{3b}+2m& =\hfill & 6m-5\hfill & 2m\text{is}\text{associated}\text{with}a\text{by}\text{addition}\text{.}\text{Undo}\text{the}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{by}\text{subtracting}2m\text{from}both\text{sides}.\hfill \\ \hfill \frac{8a}{3b}+2m-2m& =\hfill & 6m-5-2m\hfill & \hfill \\ \hfill \frac{8a}{3b}& =\hfill & 4m-5\hfill & 3bassociatedwitha\text{by}\text{division}\text{.}\text{Undo}\text{the}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{by}\text{multiplying}both\text{sides}\text{by}3b.\hfill \\ \hfill (3b)\left(\frac{8a}{3b}\right)& =\hfill & 3b(4m-5)\hfill & \hfill \\ \hfill 8a& =\hfill & 12bm-15b\hfill & 8\text{is}\text{associated}\text{with}a\text{by}\text{multiplication}\text{.}\text{Undo}\text{the}\hfill \\ \hfill & \hfill & \hfill & \text{multiplication}\text{by}\text{dividing}both\text{sides}\text{by}8.\hfill \\ \hfill \frac{8a}{8}& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \\ \hfill a& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \end{array}$

Solve $4x+1-3x=(-2)(4)$ for $x.$

$\begin{array}{lll}\hfill 4x+1-3x& =\hfill & (-2)(4)\hfill \\ \hfill x+1& =\hfill & -8\hfill \\ \hfill x& =\hfill & -9\hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 4(-9)+1-3(-9)& =\hfill & -8\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill & \hfill -36+1+27& =\hfill & -8\hfill & \text{Is}\text{this}\text{correct?}\hfill \\ \hfill & \hfill -8& =\hfill & -8\hfill & \text{Yes,}\text{this}\text{is}\text{correct}\text{.}\hfill \end{array}$