Step by step
The question
Question: solve y = 2x − 1 and 3x + y = 14.
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.
1 Put ① into ②
In ①, y is already alone.
y is the same as 2x − 1.
So in ②, we write (2x − 1) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
2 Find x
A bracket with a plus or nothing in front can just be removed.
All the x terms go together.
3x + 2x = 5x.
Now we have 5x − 1 = 14.
We want x by itself.
The − 1 is in the way.
Is x alone now?
Not yet: − 1 is still with x.
How do we remove it?
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 cancel each other.
They make zero.
14 plus 1 is 15.
Now we have 5x = 15.
5x means 5 times x.
Is x alone now?
Not yet: 5x means 5 times x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
They cancel each other.
So just x is left.
15 divided by 5 is 3.
So x = 3.
So after step 2: we added 1 and divided by 5, and now we have x = 3.
3 Find y
Now we put x = 3 into ①.
Times first.
That is 6 − 1.
So y = 5.
So x = 3 and y = 5, both at the same time.
We write it as (3, 5).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 3 and y = 5 into ①.
Now the right side.
We get 5.
The left side is 5 too.
Both sides are the same.
Now ②.
We get 14, the same as the right side.
Both work.
So the answer is right!
5 Pick the choice
Our answer is (x, y) = (3, 5).
That is choice A.
6 Why not the others
B is (3, 7).
Wrong: y = 2 × 3 + 1: added 1 instead of taking it away.
C is (5, 3).
Wrong: x and y swapped.
D is (15, 29).
Wrong: 5x = 15 means x = 3: forgot to divide by 5.
Let's look back.
First: put ① into ②.
Then: find x.
Then: find y.
Then: check.
Then: pick the choice.
Last: why not the others.