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=y2−y1x2−x1
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=2−4−7−−2=2−4−7+2=−2−5=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 -5/2−52. The point-slope form of a linear equation is: (y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))(y−y1)=m(x−x1)
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))(y−3)=−52(x−−1)
(y - color(blue)(3)) = color(red)(-5/2)(x + color(blue)(1))(y−3)=−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))y−3=(−52×x)+(−52×1)
y - color(blue)(3) = -5/2x - 5/2y−3=−52x−52
y - color(blue)(3) + 3 = -5/2x - 5/2 + 3y−3+3=−52x−52+3
y - 0 = -5/2x - 5/2 + (2/2 xx 3)y−0=−52x−52+(22×3)
y = -5/2x - 5/2 + 6/2y=−52x−52+62
y = -5/2x + 1/2y=−52x+12