First, we need to find the slope of the line which passes through (-2, 4) and (-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))
Where m is the slope and (color(blue)(x_1, y_1)) and (color(red)(x_2, y_2)) 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 = 2/5
A perpendicular slope is the negative inverse of the original slope. Let's call the perpendicular slope m_p.
The we can say: m_p = -1/m
Or, for this problem:
m_p = -1/(2/5) = -5/2
We can now use the point-slope formula to find the equation of the line passing through (-1, 3) with a slope of -5/2. The point-slope form of a linear equation is: (y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))
Where (color(blue)(x_1), color(blue)(y_1)) is a point on the line and color(red)(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 - color(blue)(3)) = color(red)(-5/2)(x + color(blue)(1))
If we want this slope-intercept form we can solve for y giving:
y - color(blue)(3) = (color(red)(-5/2) xx x) + (color(red)(-5/2) xx color(blue)(1))
y - color(blue)(3) = -5/2x - 5/2
y - color(blue)(3) + 3 = -5/2x - 5/2 + 3
y - 0 = -5/2x - 5/2 + (2/2 xx 3)
y = -5/2x - 5/2 + 6/2
y = -5/2x + 1/2
