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