There are two ways of using information to write the equation of a line. Both work equally as well, and unless are you asked to use a specific method, you can use whichever one makes more sense to you.

Method One:Slope-Intercept Form (\(y = mx + b\)) and solving for “b”.

Example 1:

Write the equation of the line with a slope of \(\Large \frac{9}{4}\) and goes through the point (4,5).

We want to get our equation in the form \(y = mx + b\). “m” stands for slope, so we have that part already! Since they also gave us a point, we can use these numbers and plug them in for x and y.

\(m = \Large \frac{9}{4}\)       \(x = 4\)        \(y = 5\)

\(y = mx + b\)

    \({\text{5 }} = \Large \frac{9}{4}\left( {\text{4}} \right){\text{ }} + b\)

Simplify and solve for b.

\({\text{5 }} = {\text{ 9 }} + b\)

\( - {\text{4 }} = b\)

Now, plug “m” and “b” into your slope-intercept form

Answer: \(y = \Large \frac{9}{4}x--{\text{ 4}}\)

Method Two:Point-Slope Form

This way might be a little less confusing, but you have to remember a formula in order to use this method.

Point-Slope Form ►\((y - {y_1}) = m(x - {x_1})\)

Example 2:

Write the equation of a line with a slope of \(\Large \frac{5}{3}\) and goes through point (3,1)

We don’t have to worry about “b” in this method, so we already have all the parts that we need!

m = \(\Large \frac{5}{3}\)           \({x_1} = 3\)            \({y_1} = 1\)

\((y - {y_1}) = m(x - {x_1})\)                  Plug in values.            

  \((y - 1) = {\Large \frac{5}{3}}(x - 3)\)                  Distributive.

\(y - 1 = {\Large \frac{5}{3}x} - 5\)               Add 1 to both sides.

Answer: \(y = {\Large \frac{5}{3}x} - 4\)

You can also figure out the equation of a line if you are only given 2 points. There is one extra step and then you can choose one of the above methods to complete it. Here’s an example:

Example 3:

Write the equation of a line that passes through points (4, -2) & (-6, 0)

It doesn’t matter which method we would like to use, for both of them we first need a slope! You can use both points to find the slope.

Slope = \({\Large \frac{{{y_2} - {y_1}}}{{{x_2} - x}}_1}\)

Let’s label our points:

\((4, - 2)( - 6,0)\)

\(({x_1},{y_1})({x_2},{y_2})\)

Now, plug them in.

\({\Large \frac{{{y_2} - {y_1}}}{{{x_2} - x}}_1} = \Large \frac{{0 - ( - 2)}}{{ - 6 - 4}} = \Large \frac{2}{{ - 10}} =  - \Large \frac{1}{5}\)

So, the slope is

\( - \Large \frac{1}{5}\)

We will finish this problem using both methods, so you can decide which way you like best.

Slope-Intercept Method

Choose one of the points that we started with. (-6,0)

\(m =  - \Large \frac{1}{5}\)        \(x =  - 6\)          \(y = 0\)

\(y = mx + b\)

\(0 =  - {\Large \frac{1}{5}}( - 6) + b\)

\(0 = {\Large \frac{6}{5}} + b\)

\( - \Large \frac{6}{5} = b\)

Answer: \(y =  - \Large \frac{1}{5}x - \Large \frac{6}{5}\)

Point-Slope Method

\(m =  - \Large \frac{1}{5}\)        \({x_1} =  - 6\)        \({y_1} = 0\)

  \((y - {y_1}) = m(x - {x_1})\)

\((y - 0) =  - \Large \frac{1}{5}(x + 6)\)

\(y =  - \Large \frac{1}{5}x - \Large \frac{6}{5}\)

Below you can download some free math worksheets and practice.