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)(-1) - color(blue)(2))/(color(red)(4) - color(blue)(-1)) = (color(red)(-1) - color(blue)(2))/(color(red)(4) + color(blue)(1)) = -3/5#
We can now use the slope we calculated and the values from the first point in the problem to write an equation in point-slope form. 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.
#(y - color(blue)(2)) = color(red)(-3/5)(x - color(blue)(-1))#
#(y - color(blue)(2)) = color(red)(-3/5)(x + color(blue)(1))#
We can also use the slope we calculated and the values from the second point in the problem to write an equation in point-slope form.
#(y - color(blue)(-1)) = color(red)(-3/5)(x - color(blue)(4))#
#(y + color(blue)(1)) = color(red)(-3/5)(x - color(blue)(4))#
We can solve this equation for #y# to write an equation is 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)(1) = -3/5x - (-3/5 xx color(blue)(4))#
#y + color(blue)(1) = -3/5x + 12/5#
#y + color(blue)(1) - 1 = -3/5x + 12/5 - 1#
#y + 0 = -3/5x + 12/5 - 5/5#
#y = color(red)(-3/5)x + color(blue)(7/5)#
We can now convert this equation to Standard Linear form. The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)#
Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
#3/5x + y = 3/5x + color(red)(-3/5)x + color(blue)(7/5)#
#3/5x + y = 0 + color(blue)(7/5)#
#3/5x + y = color(blue)(7/5)#
#5(3/5x + y) = 5 xx color(blue)(7/5)#
#color(red)(3)x + color(blue)(5)y = color(green)(7)#
