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)(3) - color(blue)(1))/(color(red)(9) - color(blue)(3)) = 2/6 = 1/3#
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 calculate and the first point from the problem gives:
#(y - color(red)(1)) = color(blue)(1/3)(x - color(red)(3))#
We can also substitute the slope we calculate and the second point from the problem giving:
#(y - color(red)(3)) = color(blue)(1/3)(x - color(red)(9))#
We can also solve for 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.
Solving our second equation for #y# gives:
#y - color(red)(3) = (color(blue)(1/3) xx x) - (color(blue)(1/3) xx color(red)(9))#
#y - color(red)(3) = 1/3x - 9/3#
#y - color(red)(3) = 1/3x - 3#
#y - color(red)(3) + 3 = 1/3x - 3 + 3#
#y = color(red)(1/3)x + color(blue)(0)# or #y = 1/3 x#
Four equations which solve this problem are:
#(y - color(red)(1)) = color(blue)(1/3)(x - color(red)(3))# or
#(y - color(red)(3)) = color(blue)(1/3)(x - color(red)(9))# or
#y = color(red)(1/3)x + color(blue)(0)# or #y = color(red)(1/3)x#
