How do you find a equation of the line containing the given pair of points (3,1) and (9,3)?

1 Answer
Feb 14, 2017

See the entire solution process below:

Explanation:

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#