What is the equation of the line that passes through (-1,3)(1,3) and is perpendicular to the line that passes through the following points: (- 2,4),(-7,2)(2,4),(7,2)?

1 Answer
Nov 12, 2017

See a solution process below:

Explanation:

First, we need to find the slope of the line which passes through (-2, 4)(2,4) and (-7, 2)(7,2). The slope can be found by using the formula: m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))m=y2y1x2x1

Where mm is the slope and (color(blue)(x_1, y_1)x1,y1) and (color(red)(x_2, y_2)x2,y2) are the two points on the line.

Substituting the values from the points in the problem gives:

m = (color(red)(2) - color(blue)(4))/(color(red)(-7) - color(blue)(-2)) = (color(red)(2) - color(blue)(4))/(color(red)(-7) + color(blue)(2)) = (-2)/-5 = 2/5m=2472=247+2=25=25

A perpendicular slope is the negative inverse of the original slope. Let's call the perpendicular slope m_pmp.

The we can say: m_p = -1/mmp=1m

Or, for this problem:

m_p = -1/(2/5) = -5/2mp=125=52

We can now use the point-slope formula to find the equation of the line passing through (-1, 3)(1,3) with a slope of -5/252. The point-slope form of a linear equation is: (y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))(yy1)=m(xx1)

Where (color(blue)(x_1), color(blue)(y_1))(x1,y1) is a point on the line and color(red)(m)m is the slope.

Substituting the slope we calculated and the values from the point in the problem gives:

(y - color(blue)(3)) = color(red)(-5/2)(x - color(blue)(-1))(y3)=52(x1)

(y - color(blue)(3)) = color(red)(-5/2)(x + color(blue)(1))(y3)=52(x+1)

If we want this slope-intercept form we can solve for yy giving:

y - color(blue)(3) = (color(red)(-5/2) xx x) + (color(red)(-5/2) xx color(blue)(1))y3=(52×x)+(52×1)

y - color(blue)(3) = -5/2x - 5/2y3=52x52

y - color(blue)(3) + 3 = -5/2x - 5/2 + 3y3+3=52x52+3

y - 0 = -5/2x - 5/2 + (2/2 xx 3)y0=52x52+(22×3)

y = -5/2x - 5/2 + 6/2y=52x52+62

y = -5/2x + 1/2y=52x+12