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 Question from Karen: Find an equation for the line with the given properties. Perpendicular to the line 7x + 2y = -6; containing the point (-2, -1) Find an equation for the line with the given properties. Parallel to the line 5x - 3y =-6; x intercept = 3

Hi Karen,

In both of your problems you have an equation of the form $ax + by = c$ where $a, b$ and $c$ are constants. The first step is to solve the equation for $y.$ I am going to look at an example $4x + 5y = 6.$ To solve for $y$ I would first add $-4x$ to each side and then divide both sides by $5.$ This yields

$y = - \frac{4}{5} x + \frac{6}{5}.$

This is in the form that your textbook probably writes

$y = mx + b.$

In this form $m$ is the slope of the line and hence the line $4x + 5y = 6$ has slope $- \large \frac4{5}.$

If you are looking for a line parallel to this line it must also have a slope of $- \large \frac4{5}$ and hence its equation has the form

$y = - \frac{4}{5} x + b$

for some constant $b.$ Use any other facts you are given to evaluate $b.$

If you are looking for a line perpendicular to $4x + 5y = 6$ this line it must also have a slope which is the negative reciprocal of the slope of $4x + 5y = 6$ and hence its slope is $\large \frac5{4}$ and hence its equation has the form

$y = \frac{5}{4} x + b$

for some constant $b.$ Use any other facts you are given to evaluate $b.$

Write back if you need more assistance,
Penny

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