Step by step
The question
Question: find the equation of the line through (2, 1) and (4, 7).
There are four choices.
First we solve it, the normal way.
Then we pick.
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, 7).
The first point is (2, 1).
7 minus 1 is 6.
4 minus 2 is 2.
6 divided by 2 is 3.
So m = 3.
2 Find b
In y = mx + b, put m = 3.
We can use either point.
We take (2, 1).
The point (2, 1) is on the line, so its x and y fit.
We put x = 2 and y = 1 in.
3 times 2 is 6.
Is b alone now?
Not yet: + 6 is still with b.
How do we remove it?
We want b by itself.
The + 6 is in the way.
To remove + 6, we take away 6.
We do it on both sides, so they stay equal.
6 and − 6 cancel each other.
They make zero.
1 minus 6 is −5.
So b = −5.
So after step 2: we took away 6, and now we have −5 = b.
3 Write the equation
Put m = 3 and b = −5 into y = mx + b.
So the line is y = 3x − 5.
4 Check
Let's check: put x = 2 in.
We get 1, the y of (2, 1).
Now put x = 4 in.
We get 7, the y of (4, 7).
Let's see both points on the grid.
Both are on the line y = 3x − 5.
Go right 1,
up 3: the slope is 3.
And it crosses the y-axis at −5, just like b says.
5 Pick the choice
Our answer is y = 3x − 5.
That is choice A.
6 Why not the others
B is y = 3x + 1.
Wrong: used b = 1, the y of (2, 1): b is y at x = 0.
C is y = 3x + 5.
Wrong: 1 − 6 is −5, not 5.
D is y = 6x − 11.
Wrong: 6/2 is 3: forgot to divide by the run.
Let's look back.
First: find the slope.
Then: find b.
Then: write the equation.
Then: check.
Then: pick the choice.
Last: why not the others.