First, determine the slope of the 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 points in the problem gives:
#m = (color(red)(2) - color(blue)(-6))/(color(red)(-3) - color(blue)(3)) = (color(red)(2) + color(blue)(6))/(color(red)(-3) - color(blue)(3)) =8/-6 = -4/3#
We can now use the point-slope formula to write an equation. 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 slope we calculated and the values from the first point in the problem gives:
#(y - color(red)(-6)) = color(blue)(-4/3)(x - color(red)(3))#
#(y + color(red)(6)) = color(blue)(-4/3)(x - color(red)(3))#
We can write another equation in point-slope form by substituting the slope we calculated and the values from the second point in the problem giving:
#(y - color(red)(2)) = color(blue)(-4/3)(x - color(red)(-3))#
#(y - color(red)(2)) = color(blue)(-4/3)(x + color(red)(3))#
We can solve this equation for #y# to transform the equation 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.
#y - color(red)(2) = (color(blue)(-4/3) * x) + (color(blue)(-4/3) * color(red)(3))#
#y - color(red)(2) = -4/3x + (-4)#
#y - color(red)(2) = -4/3x - 4#
#y - color(red)(2) + 2 = -4/3x - 4 + 2#
#y - 0 = -4/3x - 2#
#y = color(red)(-4/3)x - color(blue)(2)#
