Finding Equations of Lines

Example 1:

Find the equation of the line through (0, 3) with slope ½.

This is the easy type because all you have to do is plug in the slope and y-intercept into the equation y = mx + b where the number in front of the x is always the slope and the b is always the y-intercept (where the line crosses the y-axis). Thus the equation of the line we are looking for is y = ½ x + 3.

If you are asked to find the equation by looking at the graph, the easiest way is usually to find the y-intercept, count the slope (rise over run), and plug the numbers in just like example 1.

To find the equation of a line you always need the slope and some point on the line. No matter what information they give you, somehow you must figure out these two things.

If they don’t just give them to you like in example 1, you can use the point-slope formula for finding the equation of a line.

m is the slope, (x₁, y₂) is any point on the line.

The problems differ only in the information they give you. From that information you can always figure out the slope and y-intercept to plug into the equation.

Example 2:

Find the equation of the line through (1, 3) with slope -2.

This differs from example 1 because the point they gave is not on the y-axis. Use the point-slope formula to get

y – 3 = -2(x – 1) Simplifying gives y – 3 = -2x + 1 and y = -2x + 1 in slope-intercept form.

Sometimes they give two points and you have to find the slope yourself. The slope can be found from two points by making a fraction where the difference in the two y-values is on the top and the difference in the two x-values is on the bottom…hence, slope is said to be rise (coming from the y- or vertical axis) over run (coming from the x- or horizontal axis).

Example 3:

Find the equation of the line passing through the points (2,3) and (1, 5).

Remember the comment above:

We have a point (actually we have two) so we need to find the slope.

Subtracting the y’s from the two points we have gives a difference of 5-3 = 2.

Subtracting the x’s from the two points we have gives a difference of 1-2 = -1.

So the slope is 2 over –1 or –2. (NOTE that you can subtract the y’s in any order you want, but whichever y you put first determines the x that must be first when you subtract the x’s. Failing to do this will make you miss the sign on the slope and thus miss the problem.

Now, we have a slope of –2 and the line goes through the point (2,3) and the problem becomes like example 2 above.

y – 3 = -2(x – 2) which simplifies to y - 3 = -2x + 4 or y = -2x + 7 .