First, we need to determine the slop 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)(3) - color(blue)(6))/(color(red)(-7) - color(blue)(5)) = (-3)/-12 = 1/4#
Next we can use the point-slope formula to write and 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)(6)) = color(blue)(1/4)(x - color(red)(5))#
We can also substitute the slope we calculated and the values from the second point in the problem giving:
#(y - color(red)(3)) = color(blue)(1/4)(x - color(red)(-7))#
#(y - color(red)(3)) = color(blue)(1/4)(x + color(red)(7))#
We can also solve this equation for #y# to put the equation in slope-intercept form. The slope-intercept form of a linear equation is: v
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y - color(red)(3) = (color(blue)(1/4) xx x) + (color(blue)(1/4) xx color(red)(7))#
#y - color(red)(3) = 1/4x + 7/4#
#y - color(red)(3) + 3 = 1/4x + 7/4 + 3#
#y = 1/4x + 7/4 + (4/4 xx 3)#
#y = 1/4x + 7/4 + 12/4#
#y = color(red)(1/4)x + color(blue)(19/4)#
