Find an equation for the line perpendicular to the line −2x+4y=2 having the same y-intercept as 4x−7y=42.

Question

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Answer 1

-2x+4y=2    is the same line as    x-2y=-1   [note: this is how it is written by convention]

All lines parallel to  x-2y=-1  have the formula  x-2y=C (where C≠-1).

 

Written in slope-intercept form (y=mx+b), the line  x-2y=-1  is  y=(1/2)x -1/2.  It has slope of m=1/2 and so do all parallel lines.

 

 

Now, perpendicular lines have slopes that are the negative reciprocal of each other [that is, m and (-1/m)].

The line y=(1/2)x -1/2 is perpendicular to lines that have formulas of the form  y=-2x+b.

The line  x-2y=-1 is perpendicular to lines that have formulas of the form  2x+y=C.

 

The line  x-2y=-1 [or y=(1/2)x-1/2] has a y-intercept [that is, the point where x=0, or (0,b)] of (-1/2).

However, the line 4x-7y=42 [or y=(4/7)x-6] has a y-intercept of (-6).

 

So, the perpendicular line to -2x+4y=2 [or x-2y=-1 or y=(1/2)x-1/2] having the same y-intercept as the line 4x-7y=42 [or y=(4/7)x-6] is:

        y=-2x-6 

    or

        2x+y=-6

 

Answer 2

Hi Monica, for a line to be perpendicular to -2x + 4y = 2, it will have a slope that is the negative reciprocal of the slope of the given line.

 

Slope of -2x + 4y = 2, in standard form, is equal to -2/-4, or 1/2.

 

The negative reciprocal of 1/2 is -2.

 

The y intercept of the line 4x - 7y = 42, is found by setting x = 0 and solving for y. 

 

0 - 7y = 42

 

y = -6

 

The y intercept point is (0, -6).

 

Use the point-slope form of the line given the slope of -2 and the point (0, -6) to find the line that is perpendicular to the given line.

 

Point-slope form of the line:

 

y - y₁ = m * (x - x₁)

 

Substitute:

 

m = -2

 

x₁ = 0

 

y₁ = -6

 

y - (-6) = -2 * (x - 0)

 

y + 6 = -2x

 

in slope-intercept form, the line is: y = -2x - 6.

 

Questions?