First, we must 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)(7) - color(blue)(1))/(color(red)(3) - color(blue)(-2)) = (color(red)(7) - color(blue)(1))/(color(red)(3) + color(blue)(2)) = 6/5#
We can now use 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 slope we calculated and the values from the first point in the problem gives:
#(y - color(red)(1)) = color(blue)(6/5)(x - color(red)(-2))#
#(y - color(red)(1)) = color(blue)(6/5)(x + color(red)(2))#
We can also substitute the slope we calculated and the values from the second point in the problem giving:
#(y - color(red)(7)) = color(blue)(6/5)(x - color(red)(3))#
We can now solve this equation for #y# to put the equation in 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)(7) = (color(blue)(6/5) xx x) - (color(blue)(6/5) xx color(red)(3))#
#y - color(red)(7) = 6/5x - 18/5#
#y - color(red)(7) + 7 = 6/5x - 18/5 + 7#
#y - 0= 6/5x - 18/5 + (7 xx 5/5)#
#y = 6/5x - 18/5 + 35/5#
#y = color(red)(6/5)x + color(blue)(17/5)#
