The point-slope formula can be used to find this equation. However, we must first find the slope which can be found using two points on a line.
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 problem gives:
m = (color(red)(2) - color(blue)(-6))/(color(red)(4) - color(blue)(5))
m = (color(red)(2) + color(blue)(6))/(color(red)(4) - color(blue)(5))
m = 8/-1 = -8
The slope and either of the points can now be used with the point-slope formula to find an equation for the line.
The point-slope formula states: (y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))
Where color(blue)(m) is the slope and color(red)(((x_1, y_1))) is a point the line passes through.
Substituting the calculate slope and the second point gives:
(y - color(red)(2)) = color(blue)(-8)(x - color(red)(4))
Or, we can convert to the more familiar slope-intercept form by solving for y:
y - color(red)(2) = (color(blue)(-8) xx x) - (color(blue)(-8) xx color(red)(4))
y - 2 = -8x + 32
y - 2 + color(red)(2) = -8x + 32 + color(red)(2)
y - 0 = -8x + 34
y = -8x + 34
Or, we can use the point-slope formula and the first point to give:
(y - color(red)(-6)) = color(blue)(-8)(x - color(red)(5))
(y + color(red)(6)) = color(blue)(-8)(x - color(red)(5))
