Given (2,-2), and and y intercept 4, how do you find the equation of the line?

We have been given two points (the y-intercept is point #(0, 4)#) so we can use the point-slope formula to find the equation for the line.

First, we must find the slope which requires two points:

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 our two points gives:

#m = (color(red)(-2) - color(blue)(4))/(color(red)(2) - color(blue)(0))#

#m = (-6)/2#

#m = -3#

Now, having the slope, we can use it and one of the points to use the point-slope formula to find the equation of 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 our slope and one of the points gives:

#(y - color(red)(4)) = color(blue)(-3)(x - color(red)(0))#

We can also put it into the more familiar slope-intercept form by solving for #y#:

#y - color(red)(4) = color(blue)(-3)x#

#y - color(red)(4) + 4 = color(blue)(-3)x + 4#

#y - 0 = color(blue)(-3)x + 4#

#y = color(blue)(-3)x + 4#