Write the equation of the line that is perpendicular to the line 4y=5x−8 through the point (-5,4)

Answer 1

As I'm sure you know already, the standard equation of a line in slope intercept form is y = mx (+ or -) b, i.e., y=mx+b or y=mx-b. Note, m can also be + or -.

The given equation, 4y = 5x - 8 has all the terms to transform it into the form of the standard equation of a line.

Note, the y-term in the standard form has a leading coefficient of 1. The given equation has a leading coefficient of 4. Divide all the terms in the given equation by 4 and you'll have y = 5x/4 - 2.

⊥ : perpendicular

Now, the line ⊥ to the above standard equation of a line in slope intercept form always has a slope that's negative and the reciprocal or flip of the slope of the standard equation of a line. Simply divide 1 by the slope of the standard equation of a line (5/4) and you'll have the slope of the ⊥ line (-4/5).

Now, w/ the ⊥ slope, we use the given coordinate pt. (-5,4) and write the equation of line ⊥ to the line 4y = 5x - 8 that passes through that given pt.

We use the point slope form of the equation of a line, y - y₁ = m(x - x₁), to arrive at the line that passes through (-5,4) with the newly found slope of the ⊥ line.

y - 4 = -4/5(x - (-5)) → simplify what's in the parenthesis 1st → (x-(-5)) = (x + 5) → multiply -4/5 w/ (x+5) → -4x/5 - 4 because -4/5 ⋅ 5 = -4 since -4/5 ⋅ 5 is the same as -4/5 ⋅ 5/1 where the 5's cancel leaving -4/1 = -4 → now we have y - 4 = -4x/5 - 4 → write equation in standard equation of a line form, y = mx + or - b → add 4 to both left and right sides of equation to cancel -4 on the left side, leaving y alone → y = -4x/5 ⇒ y = -4x/5 is the equation of a line that's ⊥ to 4y = 5x - 8.

Answer 2

The slope of the perpendicular line is -4/5 (the negative reciprocal of the slope of the given line!).

The equation of the perpendicular is -4/5 = (y-4)/(x+5) and

this equation can be manipulated into any desired form.