Step by step
The question
Question: find the line through (2, −1), parallel to 2x + y = 4.
1 Slope of the given line
The given line is not y = mx + b yet.
So we write it as y = mx + b first.
We want y by itself.
The 2x is in the way.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
Write what is left.
We write the x part first, like in y = mx + b.
Each part keeps its sign.
y is by itself.
m is the number in front of x: −2.
It is the slope.
2 Slope of the new line
Parallel lines have the same slope.
So the new line has slope −2 too.
Careful!
Same slope, but not the same b.
The new line goes through (2, −1), so it has its own b.
3 Find b
In y = mx + b, put m = −2.
Then use the point (2, −1): put x = 2 and y = −1 in.
Minus times plus is minus.
−2 times 2 is −4.
We want b by itself.
The − 4 is in the way.
To remove − 4, we add 4.
We do it on both sides, so they stay equal.
−4 and + 4 make zero.
Cross them out.
−1 plus 4 is 3.
So b = 3.
4 Write the equation
Put m = −2 and b = 3 into y = mx + b.
So the new line is y = −2x + 3.
5 Check
Let's check: put x = 2 into y = −2x + 3.
We get −1, the y of (2, −1).
And the slope is −2, the same as the given line.
Parallel!
Let's see it on the grid.
The given line in blue.
Our new line y = −2x + 3 in red.
It goes through (2, −1).
They never meet.
Let's look back.
First: slope of the given line.
Then: slope of the new line.
Then: find b.
Then: write the equation.
Last: check.