First, we need to determine the slope of the line passing through the two points from the problem. 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)(11) - color(blue)(-3))/(color(red)(5) - color(blue)(2)) = (color(red)(11) + color(blue)(3))/(color(red)(5) - color(blue)(2)) = 14/3#
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)(-3)) = color(blue)(14/3)(x - color(red)(2))#
#color(green)(Solution 1))# #(y + color(red)(3)) = color(blue)(14/3)(x - color(red)(2))#
We can also substitute the slope we calculated and the second point from the problem giving:
#color(green)(Solution 2))# #(y - color(red)(11)) = color(blue)(14/3)(x - color(red)(5))#
We can also solve one of these equations for #y# to put it 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)(3) = (color(blue)(14/3) xx x) - (color(blue)(14/3) xx color(red)(2))#
#y + color(red)(3) = 14/3x - 28/3#
#y + color(red)(3) - 3 = 14/3x - 28/3 - 3#
#y + 0 = 14/3x - 28/3 - (3 xx 3/3)#
#y = 14/3x - 28/3 - 9/3#
#color(green)(Solution 3))# #y = color(red)(14/3)x - color(blue)(37/3)#
