Step by step
The question
Question: solve a + 2b = 7 and 3a − b = 7 by substitution.
We call them ① and ②.
We want the a and b that make both true.
1 Get a alone in ①
No letter is alone yet.
In ①, a has just 1 in front.
So we get a alone there first.
We want a by itself.
The + 2b is in the way.
To remove + 2b, we take away 2b.
We do it on both sides, so they stay equal.
+ 2b and − 2b make zero.
Cross them out.
Write what is left.
We write the b part first.
Each part keeps its sign.
a is by itself.
We call it ③.
2 Put ③ into ②
In ③, a is now alone.
a is the same as −2b + 7.
So in ②, we write (−2b + 7) in place of a.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find b
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Careful!
Not only the first part!
7 is multiplied by 3 too.
Work out each multiplication.
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the b terms go together.
−6b − b = −7b.
Now we have −7b + 21 = 7.
We want b by itself.
The + 21 is in the way.
To remove + 21, we take away 21.
We do it on both sides, so they stay equal.
+ 21 and − 21 make zero.
Cross them out.
7 minus 21 is −14.
Now we have −7b = −14.
−7b means −7 times b.
We want just b.
To remove times −7, we divide by −7.
We do it on both sides, so they stay equal.
−7 divided by −7 is 1.
So just b is left.
−14 divided by −7 is 2.
So b = 2.
4 Find a
Now we put b = 2 into ③.
That is −4 + 7.
So a = 3.
So a = 3 and b = 2, both at the same time.
We write it as (3, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put a = 3 and b = 2 into ①.
We get 7, the same as the right side.
Now ②.
We get 7, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get a alone in ①.
Then: put ③ into ②.
Then: find b.
Then: find a.
Last: check.