Step by step
The question
Question: find the equation of the line through (1, 4) and (4, 13).
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.
So we need m and b.
1 Find the slope
We find the slope m with the slope formula.
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
Put the numbers in.
The second point is (4, 13).
The first point is (1, 4).
13 minus 4 is 9.
4 minus 1 is 3.
9 divided by 3 is 3.
So m = 3.
2 Find b
In y = mx + b, put m = 3.
We can use either point.
We take (1, 4).
The point (1, 4) is on the line, so its x and y fit.
We put x = 1 and y = 4 in.
3 times 1 is 3.
Is b alone now?
Not yet: + 3 is still with b.
How do we remove it?
We want b by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
We do it on both sides, so they stay equal.
3 and − 3 cancel each other.
They make zero.
4 minus 3 is 1.
So b = 1.
So after step 2: we took away 3, and now we have 1 = b.
3 Write the equation
Put m = 3 and b = 1 into y = mx + b.
So the line is y = 3x + 1.
4 Check
Let's check: put x = 1 in.
We get 4, the y of (1, 4).
Now put x = 4 in.
We get 13, the y of (4, 13).
Let's see both points on the grid.
Both are on the line y = 3x + 1.
Go right 1,
up 3: the slope is 3.
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.