First, we need to determine the slope. 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)(0) - color(blue)(10))/(color(red)(9) - color(blue)(-17)) = (color(red)(0) - color(blue)(10))/(color(red)(9) + color(blue)(17)) = -10/26 = -5/13#
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)(10)) = color(blue)(-5/13)(x - color(red)(-17))#
#(y - color(red)(10)) = color(blue)(-5/13)(x + color(red)(17))#
Or, we can substitute the slope we calculated and the second point from the problem giving:
#(y - color(red)(0)) = color(blue)(-5/13)(x - color(red)(9))#
#y = color(blue)(-5/13)(x - color(red)(9))#
Or, we can expand this 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(blue)(-5/13) xx x) - (color(blue)(-5/13) xx color(red)(9))#
#y = color(red)(-5/13)x + color(blue)(45/13)#
Three possible solutions are:
#(y - color(red)(10)) = color(blue)(-5/13)(x + color(red)(17))#
#y = color(blue)(-5/13)(x - color(red)(9))#
#y = color(red)(-5/13)x + color(blue)(45/13)#
