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)(1))/(color(red)(9) - color(blue)(3)) = 4/6 = 2/3#
Next, we can use the point-slope formula to write and 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)(1)) = color(blue)(2/3)(x - color(red)(3))#
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 for #y# gives:
#y - color(red)(1) = (color(blue)(2/3) xx x) - (color(blue)(2/3) xx color(red)(3))#
#y - color(red)(1) = 2/3x - (color(blue)(2/cancel(3)) xx cancel(color(red)(3)))#
#y - color(red)(1) = 2/3x - 2#
#y - color(red)(1) + 1 = 2/3x - 2 + 1#
#y - 0 = 2/3x - 1#
#y = color(red)(2/3)x - color(blue)(1)#
