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In Sections [link] and [link] we worked with techniques that involved the use of addition, subtraction, multiplication, and division to solve equations. We can combine these techniques to solve more complicated equations. To do so, it is helpful to recall that an equation is solved for a particular variable when all other numbers and/or letters have been disassociated from it and it is alone on one side of the equal sign. We will also note that
To associate numbers and letters we use the order of operations.
To undo an association between numbers and letters we use the order of operations in reverse.
Solve $4x-7=9$ for $x.$
$\begin{array}{llll}\hfill 4x-7& =\hfill & 9\hfill & \text{First,}\text{\hspace{0.17em}}\text{undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{between}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}7.\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{adding}\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{.}\hfill \\ \hfill 4x-7+7& =\hfill & 9+7\hfill & \hfill \\ \hfill 4x& =\hfill & 16\hfill & \text{Now,}\text{\hspace{0.17em}}\text{undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{between}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}4.\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}.\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill \frac{4x}{4}& =& \frac{16}{4}& \\ \hfill 16-7& =\hfill & 9\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill x& =\hfill & 4\hfill & \hfill \end{array}$
$\begin{array}{lllll}Check:\hfill & \hfill 4(4)-7& =\hfill & 9\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 9& =\hfill & 9\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$
Solve $\frac{3y}{4}-5=-11.$
$\begin{array}{llll}\hfill \frac{3y}{4}-5& =\hfill & -11\hfill & -5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{adding}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{.}\hfill \\ \hfill \frac{3y}{4}-5+5& =\hfill & -11+5\hfill & \hfill \\ \hfill \frac{3y}{4}& =\hfill & -6\hfill & 4\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{division}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplying}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}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{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}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}$
$\begin{array}{lllll}Check:\hfill & \hfill \frac{3(-8)}{4}-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill \frac{-24}{4}-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -6-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -11& =\hfill & -11\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$
Solve $\frac{8a}{3b}+2m=6m-5$ for $a.$
$\begin{array}{llll}\hfill \frac{8a}{3b}+2m& =\hfill & 6m-5\hfill & 2m\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{addition}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtracting}\text{\hspace{0.17em}}2m\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill \frac{8a}{3b}+2m-2m& =\hfill & 6m-5-2m\hfill & \hfill \\ \hfill \frac{8a}{3b}& =\hfill & 4m-5\hfill & 3b\text{\hspace{0.17em}}\text{\hspace{0.17em}}associated\text{\hspace{0.17em}}with\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{division}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplying}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}3b.\hfill \\ \hfill (3b)\left(\frac{8a}{3b}\right)& =\hfill & 3b(4m-5)\hfill & \hfill \\ \hfill 8a& =\hfill & 12bm-15b\hfill & 8\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\hfill \\ \hfill & \hfill & \hfill & \text{multiplication}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}8.\hfill \\ \hfill \frac{8a}{8}& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \\ \hfill a& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \end{array}$
Sometimes when solving an equation it is necessary to simplify the expressions composing it.
Solve $4x+1-3x=(-2)(4)$ for $x.$
$\begin{array}{lll}\hfill 4x+1-3x& =\hfill & (-2)(4)\hfill \\ \hfill x+1& =\hfill & -8\hfill \\ \hfill x& =\hfill & -9\hfill \end{array}$
$\begin{array}{lllll}Check:\hfill & \hfill 4(-9)+1-3(-9)& =\hfill & -8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -36+1+27& =\hfill & -8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -8& =\hfill & -8\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$
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