Step by step
The question
Question: solve x + y = 10 and x − y = 4.
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 y: y and −y.
They are opposites.
Added, they make zero.
y and −y make zero.
Cross them out.
We add ① and ②: the x parts together, and the numbers together.
x + x = 2x.
10 + 4 = 14.
Now we have 2x = 14.
Is x alone now?
Not yet: 2x means 2 times x.
How do we remove it?
2 Find x
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
They cancel each other.
So just x is left.
14 divided by 2 is 7.
So x = 7.
So after step 2: we divided by 2, and now we have x = 7.
3 Find y
We put x = 7 into ①.
We want y by itself.
The 7 is in the way.
Is y alone now?
Not yet: + 7 is still with y.
To remove 7, we take away 7.
We do it on both sides, so they stay equal.
7 and − 7 cancel each other.
They make zero.
10 minus 7 is 3.
So y = 3.
So x = 7 and y = 3, both at the same time.
We write it as (7, 3).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 7 and y = 3 into ①.
We get 10, the same as the right side.
Now ②.
We get 4, the same as the right side.
Both work.
So the answer is right!
5 Pick the choice
Our answer is (x, y) = (7, 3).
That is choice C.
6 Why not the others
A is (3, 7).
Wrong: x and y swapped.
B is (7, −3).
Wrong: 7 − y = 4 gives y = 3, not −3.
D is (14, −4).
Wrong: 2x = 14 means x = 7: forgot to divide by 2.
Let's look back.
First: add the equations.
Then: find x.
Then: find y.
Then: check.
Then: pick the choice.
Last: why not the others.