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)(-5) - color(blue)(-8))/(color(red)(2) - color(blue)(3/4)) = (color(red)(-5) + color(blue)(8))/(color(red)(8/4) - color(blue)(3/4)) = 3/(5/4) = 12/5#
Now, we can use the point-slope formula to write 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 second point gives:
#(y - color(red)(-5)) = color(blue)(12/5)(x - color(red)(2))#
#(y + color(red)(5)) = color(blue)(12/5)(x - color(red)(2))#
We can also substitute the slope we calculated and the first point giving:
#(y - color(red)(-8)) = color(blue)(12/5)(x - color(red)(3/4))#
#(y + color(red)(8)) = color(blue)(12/5)(x - color(red)(3/4))#
We can also solve the first 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)(5) = (color(blue)(12/5) xx x) - (color(blue)(12/5) xx color(red)(2))#
#y + color(red)(5) = 12/5x - 24/5#
#y + color(red)(5) - 5 = 12/5x - 24/5 - 5#
#y + 0 = 12/5x - 24/5 - 25/5#
#y = color(red)(12/5)x - color(blue)(49/5)#
