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We have seen that the general form of a linear equation in two variables is $ax+by=c$ (Section [link] ). When this equation is solved for $y$ , the resulting form is called the slope-intercept form. Let's generate this new form.
$\begin{array}{rrrr}\hfill ax+by& \hfill =& c\hfill & \hfill \text{Subtract}\text{\hspace{0.17em}}ax\text{\hspace{0.17em}}\text{from both sides}\text{.}\\ \hfill by& \hfill =& -ax+c\hfill & \hfill \text{Divide}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides by}\text{\hspace{0.17em}}b\\ \hfill \frac{by}{b}& \hfill =& \frac{-ax}{b}+\frac{c}{b}\hfill & \hfill \\ \hfill \frac{\overline{)b}y}{\overline{)b}}& \hfill =& \frac{-ax}{b}+\frac{c}{b}\hfill & \hfill \\ \hfill y& \hfill =& \frac{-ax}{b}+\frac{c}{b}\hfill & \hfill \\ \hfill y& \hfill =& \frac{-ax}{b}+\frac{c}{b}\hfill & \hfill \end{array}$
This equation is of the form $y=mx+b$ if we replace $\frac{-a}{b}$ with $m$ and constant $\frac{c}{b}$ with $b$ . ( Note: The fact that we let $b=\frac{c}{b}$ is unfortunate and occurs beacuse of the letters we have chosen to use in the general form. The letter $b$ occurs on both sides of the equal sign and may not represent the same value at all. This problem is one of the historical convention and, fortunately, does not occur very often.)
The following examples illustrate this procedure.
Solve $3x+2y=6$ for $y$ .
$\begin{array}{rrrr}\hfill 3x+2y& \hfill =& 6\hfill & \hfill \text{Subtract 3}x\text{\hspace{0.17em}}\text{from both sides}\text{.}\\ \hfill 2y& \hfill =& -3x+6\hfill & \text{Divide both sides by 2}\text{.}\hfill \\ \hfill y& \hfill =& -\frac{3}{2}x+3\hfill & \hfill \end{array}$
This equation is of the form $y=mx+b$ . In this case, $m=-\frac{3}{2}$ and $b=3$ .
Solve $-15x+5y=20$ for $y$ .
$\begin{array}{rrr}\hfill -15x+5y& \hfill =& 20\hfill \\ \hfill 5y& \hfill =& \hfill 15x+20\\ \hfill y& \hfill =& 3x+4\hfill \end{array}$
This equation is of the form $y=mx+b$ . In this case, $m=3$ and $b=4$ .
Solve $4x-y=0$ for $y$ .
$\begin{array}{rrr}\hfill 4x-y& \hfill =& 0\hfill \\ \hfill -y& \hfill =& -4x\hfill \\ \hfill y& \hfill =& 4x\hfill \end{array}$
This equation is of the form $y=mx+b$ . In this case, $m=4$ and $b=0$ . Notice that we can write $y=4x$ as $y=4x+0$ .
The following equations are in slope-intercept form:
$\begin{array}{cc}y=6x-7.& \text{In}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{case}\text{\hspace{0.17em}}m=6\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}b=-7.\end{array}$
$\begin{array}{cc}y=-2x+9.& \text{In}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{case}\text{\hspace{0.17em}}m=-2\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}b=9.\end{array}$
$\begin{array}{cc}y=\frac{1}{5}x+4.8& \text{In}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{case}\text{\hspace{0.17em}}m=\frac{1}{5}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}b=\mathrm{4.8.}\end{array}$
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