Question

Find the distance between the lines with equations 4x - 3y = 8 and y = 3/4x + 4.

Answer 1

What do you mean by "distance between the lines"? The two given lines intersect. Distance can mean anything you want it to in this context.

Answer 2

I'm pretty confused myself, all I know is the answer for this type of question is represented in fraction form, e.g 4/5

Answer 3

And this is all the information you're given?

Answer 4

Yes, noted it is likely that 3/4x+4 looks like this \[\frac{ 3 }{ 4 }x +4\]

Answer 5

as opposed to 4x

Answer 6

Right, that's what I was assuming. I'm afraid your question in its current state is unanswerable unless you had a concrete meaning for "distance."

Answer 7

y = (3/4)x + 4
[When you put fractions in a one-line format, it's necessary to be clear where the numerator and denominator begin and end. Without the parentheses, it was ambiguous whether the x was in the denominator.]

The lines are, of course, parallel, and their slope is 3/4.
We measure the distance between them along a perpendicular line, which has slope
m = -1/(3.4) = -4/3

Since the second equation is in slope-intercept form, it's easy to see it contains the y-intercept, (0,4). We can use that to write a point-slope equation for a perpendicular line:
y - 4 = -(4/3) (x - 0)
3y - 12 = -4x
4x + 3y = 12

This perpendicular line intersects the first line we were given somewhere. We find that intersection by solving the system:
{1} 3x - 4y = 8 [the given line]
{2} 4x + 3y = 12 [the perpendicular to it]
{3} 9x - 12y = 24 [multiplying {1} by 3]
{4} 16x + 12y = 48 [multiplying {2} by 4]
{5} 25x = 72 [adding {3} and {4}]
{6} x = 72/25 [solving {5}]
{7} 216/25 - 4y = 8 [substituting for x in {1}]
{8} 216/25 - 200/25 = 4y [adding 4y=8 to each side of {7}]
{9} 4/25 = y [solving {8}]

We get the common solution (72/25, 4/25) as the point where our perpendicular meets the other line.
It was drawn from (0,4) on the first line, so the distance between these two points is the distance between the lines.

√[(72/25)^2 + (4 - 4/25)^2]
= √[(5184/625) + (9216/625)]
= √[14400/625]
= 120/25
= 24/5 = 4.8

Answer 8

wow thank you @theghostlycreeper the answer for this is 24/5 or should I put in decimal form. It isn't specified but which do you think is more complete