First, we need to 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)(13) - color(blue)(7))/(color(red)(-3) - color(blue)(-1)) = (color(red)(13) - color(blue)(7))/(color(red)(-3) + color(blue)(1)) = 6/-2 = -3#
Next, we can use the point slope formula to write and 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 and the values from the first point in the problem gives:
#(y - color(blue)(7)) = color(red)(-3)(x - color(blue)(-1))#
#(y - color(blue)(7)) = color(red)(-3)(x + color(blue)(1))#
We can also substitute the slope we calculated and the values from the second point in the problem giving:
#(y - color(blue)(13)) = color(red)(-3)(x - color(blue)(-3))#
#(y - color(blue)(13)) = color(red)(-3)(x + color(blue)(3))#
If necessary, we can transform this equation into 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(blue)(13) = (color(red)(-3) xx x) + (color(red)(-3) xx color(blue)(3))#
#y - color(blue)(13) = -3x + (-9)#
#y - color(blue)(13) = -3x - 9#
#y - color(blue)(13) + 13 = -3x - 9 + 13#
#y - 0 = -3x + 4#
#y = color(red)(-3)x + color(blue)(4)#
