Step by step
The question
Question: solve x + 2y = 8 and −x + y = 1.
There are four choices.
First we solve it, the normal way.
Then we pick.
We call them ① and ②.
We want the x and y that make both true.
No letter is alone here.
So we add the equations to remove one letter.
This way is called elimination.
1 Add the equations
Look at x: x and −x.
They are opposites.
Added, they make zero.
x and −x make zero.
Cross them out.
We add ① and ②: the y parts together, and the numbers together.
2y + y = 3y.
8 + 1 = 9.
Now we have 3y = 9.
Is y alone now?
Not yet: 3y means 3 times y.
How do we remove it?
2 Find y
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
They cancel each other.
So just y is left.
9 divided by 3 is 3.
So y = 3.
So after step 2: we divided by 3, and now we have y = 3.
3 Find x
We put y = 3 into ①.
2 times 3 is 6.
Now we have x + 6 = 8.
We want x by itself.
The + 6 is in the way.
Is x alone now?
Not yet: + 6 is still with x.
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.
8 minus 6 is 2.
So x = 2.
So x = 2 and y = 3, both at the same time.
We write it as (2, 3).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 2 and y = 3 into ①.
We get 8, the same as the right side.
Now ②.
We get 1, the same as the right side.
Both work.
So the answer is right!
5 Pick the choice
Our answer is (x, y) = (2, 3).
That is choice B.
6 Why not the others
A is (−2, 3).
Wrong: lost the minus sign: x = 2.
C is (3, 2).
Wrong: x and y swapped.
D is (3, 4).
Wrong: 3y = 9 means y = 3: forgot to divide by 3.
Let's look back.
First: add the equations.
Then: find y.
Then: find x.
Then: check.
Then: pick the choice.
Last: why not the others.