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# Write the equation $y + 2 = \frac { 1 } { 4 } ( x + 3 )$ in the following forms: a. Slope-intercept form b. Standard form c. Standard form with a positive x-term d. Standard form with no fractions or decimals Solution

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a. Slope-intercept form is $y=mx+b$ so solve the equation for $y$ to write it in slope-intercept form:

\begin{align*} y+2&=\frac{1}{4}(x+3)\\ y+2&=\frac{1}{4}x+\frac{3}{4}&&\text{Distribute.}\\ y&=\frac{1}{4}x+\frac{3}{4}-2&&\text{Subtract 2 on both sides.}\\ y&=\frac{1}{4}x+\frac{3}{4}-\frac{8}{4}&&\text{Get a common denominator.}\\ y&=\frac{1}{4}x-\frac{5}{4}&&\text{Subtract the fractions.} \end{align*}

b. Standard form is $Ax+By=C$ so move the $x$-term of the slope-intercept form to the left side:

\begin{align*} y&=\frac{1}{4}x-\frac{5}{4}&&\text{Slope-intercept form.}\\ -\frac{1}{4}x+y&=-\frac{5}{4}&&\text{Subtract \frac{1}{4}x on both sides.} \end{align*}

c. Multiply both sides of the standard form from part b by $-1$ to write the standard form with a positive $x$-term:

\begin{align*} -\frac{1}{4}x+y&=-\frac{5}{4}&&\text{Standard form from part b.}\\ -1\left(-\frac{1}{4}x+y\right)&=-1\left(-\frac{5}{4}\right)&&\text{Multiply both sides by -1.}\\ \frac{1}{4}x-y&=\frac{5}{4}&&\text{Distribute and multiply.} \end{align*}

d. Multiply both sides of the standard form from part c by 4 to eliminate the fractions:

\begin{align*} \frac{1}{4}x-y&=\frac{5}{4}&&\text{Standard form from part c.}\\ 4\left(\frac{1}{4}x-y\right)&=4\left(\frac{5}{4}\right)&&\text{Multiply both sides by 4.}\\ x-4y&=5&&\text{Distribute and multiply.} \end{align*}

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