Line (1, 1) (2, 3) Step by Step

Step by step

The question

Question: find the equation of the line through (1, 1) and (2, 3).

(1, 1) and (2, 3)

Every line that is not straight up and down is y = mx + b.

m is the slope.

b is where the line crosses the y-axis: where we start.

So we need m and b.

y = mx + b

m = slope

b = start

1 Find the slope

We call the slope m.

m = rise over run.

m = riserun

The top is the change in y: second y minus first y.

The bottom is the change in x: second x minus first x.

m = y₂ − y₁x₂ − x₁

Put the numbers in.

The second point is (2, 3).

The first point is (1, 1).

m = 3 − 12 − 1

3 minus 1 is 2.

2 minus 1 is 1.

m = 21

2 divided by 1 is 2.

So m = 2.

m = 2

2 Find b

In y = mx + b, put m = 2.

Then use the point (1, 1): put x = 1 and y = 1 in.

1 = 2 × 1 + b

2 times 1 is 2.

1 = 2 + b

We want b by itself.

The 2 is in the way.

To remove 2, we take away 2.

We do it on both sides, so they stay equal.

1 − 2 = 2 + b ~~− 2~~

2 and − 2 make zero.

Cross them out.

1 minus 2 is −1.

So b = −1.

−1 = b

3 Write the equation

Put m = 2 and b = −1 into y = mx + b.

So the line is y = 2x − 1.

y = 2x − 1

4 Check

Let's check: put x = 1 in.

We get 1, the y of (1, 1).

y = 2(1) − 1 = 2 − 1 = 1

Now x = 2 in.

We get 3, the y of (2, 3).

y = 2(2) − 1 = 4 − 1 = 3

Let's see it.

Both points are on the line y = 2x − 1.

And it crosses the y-axis at −1, just like b says.

Let's look back.

First: find the slope.

Then: find b.

Then: write the equation.

Last: check.

Answer