How do you find the equation of the line that goes through #(- 4,3)# and #( 5,- 2)#?

First, we need to determine the slope of the line. The formula for find the slope of a line is:

#m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #(color(blue)(x_1), color(blue)(y_1))# and #(color(red)(x_2), color(red)(y_2))# are two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(-2) - color(blue)(3))/(color(red)(5) - color(blue)(-4)) = (color(red)(-2) - color(blue)(3))/(color(red)(5) + color(blue)(4)) = -5/9#

Now, we can use the point-slope formula to write an equation for the line. 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 above and the values from the first point in the problem gives:

#(y - color(blue)(3)) = color(red)(-5/9)(x - color(blue)(-4))#

#(y - color(blue)(3)) = color(red)(-5/9)(x + color(blue)(4))#

We can also substitute the slope we calculated above and the values from the second point in the problem giving:

#(y - color(blue)(-2)) = color(red)(-5/9)(x - color(blue)(5))#

#(y + color(blue)(2)) = color(red)(-5/9)(x - color(blue)(5))#