Consider the line -5x+4y=2

Question

Find the equation of the line that is parallel to this line and passes through the point ( 5,4).

Find the equation of the line that is perpendicular to this line and passes through the point (5 ,4).

Answer 1

Parallel line:

1) Convert the line given into slope-intercept form (y=mx+b)

2) Keep the m value (lines with the same m value are parallel), replace the x with 5 and the y with 4 (your given ordered pair), and replace the b value (1/2) as the variable b.

3) Solve for the variable b.

4) Re-write the new equation with your m value and the b value you solved for - x and y become variables again.

Perpendicular line:

Same process except instead of keeping the m value, use the reciprocal of the m value (if the m value is 5/4, the reciprocal is -4/5). Perpendicular lines have reciprocal (opposite) slopes!

Answer 2

Hi. So you need the equation of a line that is parallel to 4y=5x+2 at (5,4).

So, first you need to put the equation into the form y = mx+b.

Therefore, y = 1/4(5x+2) or y = (5/4)x + 1/2

So now you know that the slope of the line (m) is 5/4.

So, if you want the equation of the line parallel to this line, then it must have the same slope (m).

So if you substitute it into the y = mx + b equation, you get:

y = (5/4)x + b

Now, since you need the equation of the line with slope 5/4 which passes thru (5,4), then you must now solve for b. So, you can substitute (5,4) for (x,y) in the equation and solve for b.

4 = (5/4)(5) + b

4 = 25/4 + b

b = 16/4 - 25/4 = -9/4

Therefore, the equation of the line parallel to the original line, passing through the point (5,4) is:

y = (5/4)x - 9/4

Now, to find the equation of the line perpendicular to the original line, you need to find the line with the slope which is the opposite reciprocal of the original line. Since the original line had the slope 5/4, the slope perpendicular to this line would be -4/5.

So, now you can substitute the slope (m) of -4/5 into the y = mx + b equation.

y = (-4/5)x + b

Now, since this line also passes through the point (5,4), you can substitute for x and y into the equation and solve again for b.

4 = (-4/5)(5) + b

4 = -4 + b

b = 8

So the equation of the line perpendicular to 4y = 5x +2, and passing through (5,4) is:

y = (-4/5)x + 8