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)(8) - color(blue)(-2))/(color(red)(-3) - color(blue)(-5)) = (color(red)(8) + color(blue)(2))/(color(red)(-3) + color(blue)(5)) = 10/2 = 5#
Now, we can 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 first point from the problem gives:
#(y - color(red)(-2)) = color(blue)(5)(x - color(red)(-5))#
#(y + color(red)(2)) = color(blue)(5)(x + color(red)(5))#
Or, we can substitute the slope we calculated and the second point from the problem giving:
#(y - color(red)(8)) = color(blue)(5)(x - color(red)(-3))#
#(y - color(red)(8)) = color(blue)(5)(x + color(red)(3))#
Or, we can solve for #y# and put the equation into the 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)(8) = (color(blue)(5) xx x) + (color(blue)(5) xx color(red)(3))#
#y - color(red)(8) = 5x + 15#
#y - color(red)(8) + 8 = 5x + 15 + 8#
#y - 0 = 5x + 23#
#y = color(red)(5)x + color(blue)(23)#
