Question

Write a slope-intercept equation for a line passing through the point (4, -3) that is parallel to the line 4x + 5y = 9. Then write a second
equation for a line passing through the point (4, - 3) that is perpendicular to the line 4x + 5y = 9

Answer 1

Answer:

Parallel Line: $y=-\frac{4}{5}x+\frac{1}{5}$

Perpendicular Line: $y=\frac{5}{4}x-8$

Step-by-step explanation:

Perpendicular Lines:

Perpendicular lines have a slope that is the negative reciprocal of each other. So for example if one line has a slope of: $\frac{a}{b}$, where "a" and "b" are just some constants, then a perpendicular line would have a slope of: $-\frac{b}{a}$, so the slope fraction is just flipped and it's the opposite sign.

Parallel Lines:

Parallel lines never intersect meaning they will have the same slope, since as one goes to the right one and up or down some amount, the only way for the other line to never intersect is to also go up or down by the same amount, which is essentially the slope. The only other condition is that parallel lines must have different y-intercepts, otherwise they have the same slope and y-intercepts meaning they're just the same line and instead of never intersecting they actually intersect at infinitely many points.

Slope-Intercept Form:

The slope-intercept form is especially useful as we can simply look at it and determine the slope and y-intercept, hence the name slope-intercept form. We want to convert into this form to determine the slope of the equation given to us so we can find the perpendicular and parallel lines.

Solving the Problem:

We're given the equation in standard form:

$4x+5y=9$

We want to convert this into slope-intercept form, which we can do by isolating the y variable. We want to get rid of the 4x term on the left side by subtracting 4x, and we want to do this to both sides to maintain equality:

$4x-4x+5y=9-4x$

Simplify:
$5y=-4x+9$

Now we want to get rid of the coefficient of 5 from the left side, and this coefficient is just multiplication so to cancel out multiplication, we divide, specifically by 5 in this case, and on both sides to maintain equality:

$\frac{5y}{5}=\frac{-4x+9}{5}$

Simplify:
$y=-\frac{4}{5}x+\frac{9}{5}$

Now we know the slope is: $-\frac{4}{5}$

Finding a Perpendicular Line:

For this we simplify find the negative reciprocal so we flip the fraction and "flip" the sign, or in other words if it's negative it becomes positive, if it's positive it becomes negative so: $-\frac{4}{5}\to \frac{5}{4}$

We can represent the general equation of a perpendicular line in slope-intercept form: $y=\frac{5}{4}x+b$, where "b" is some constant number. We're given that it passes through the point (4, -3) which we can plug in as (x, y) to solve for that value "b"

$-3=\frac{5}{4}(4)+b$

Simplify on the right side

$-3=5+b$

Subtract 5 from both sides

$-8=b$

So now we have the perpendicular line equation: $y=\frac{5}{4}x-8$

Finding a Parallel Line:

For this we don't have to do anything to the slope, since for parallel lines the slope is the same, so we know the slope is: $-\frac{4}{5}$ and we can plug this into the slope-intercept form to get a general equation for a parallel line: $y=-\frac{4}{5}x+b\text{ where }b\ne\frac{9}{5}$

Since we're given that it passes through the point (4, -3) we can plug this in as (x, y) to find a definitive equation.

$-3=-\frac{4}{5}(4)+b$

Simplify on the right side:

$-3=-\frac{16}{5}+b$

Rewrite the left side to have a denominator of 5:

$-\frac{15}{5}=-\frac{16}{5}+b$

Add 16/5 to both sides:

$\frac{1}{5}=b$

So now let's plug this into a general equation we had above to get:
$y=-\frac{4}{5}x+\frac{1}{5}$