First, we will write and equation in point-slope form and then convert to slope-intercept form.
To use the point-slope form we must first determine the slope.
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)(3) - color(blue)(-3))/(color(red)(2) - color(blue)(4))
m = (color(red)(3) + color(blue)(3))/(color(red)(2) - color(blue)(4))
m = 6/-2 = -3
We can now use this calculated slope and either point to write the equation in point-slope form.
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.
Again, substituting gives:
(y - color(red)(-3)) = color(blue)(-3)(x - color(red)(4))
(y + color(red)(3)) = color(blue)(-3)(x - color(red)(4))
We can now convert this to slope-intercept form.
The slope-intercept form of a linear equation is:
y = color(red)(m)x + color(blue)(b)
Where color(red)(m) is the slope and color(blue)(b) is the y-intercept value.
We can solve our equation for y:
(y + color(red)(3)) = color(blue)(-3)(x - color(red)(4))
y + color(red)(3) = (color(blue)(-3) xx x) - (color(blue)(-3) xx color(red)(4))
y + 3 = -3x - (-12)
y + 3 = -3x + 12
y + 3 - color(red)(3) = -3x + 12 - color(red)(3)
y + 0 = -3x + 9
y = -3x + 9