Worked examples
The question
Question: solve 2x + y = 7 and x − 2y = 6 by substitution.
2x + y = 7①
We call them ① and ②.
We want the x and y that make both true.
x − 2y = 6②
1 Get y alone in ①
No letter is alone yet.
In ①, y has just 1 in front.
So we get y alone there 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 + y − 2x = 7 − 2x
2x and − 2x make zero.
Cross them out.
Write what is left.
y = 7 − 2x
We write the x part first.
Each part keeps its sign.
y is by itself.
We call it ③.
y = −2x + 7③
2 Put ③ into ②
In ③, y is now alone.
y is the same as −2x + 7.
So in ②, we write (−2x + 7) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
x − 2(−2x + 7) = 6
3 Find x
The −2 in front of the bracket multiplies every part inside.
So we write −2 times each part.
x − 2 × (−2x) − 2 × 7 = 6
Careful!
Not only the first part!
7 is multiplied by −2 too.
Careful: not 4x + 7 ✗
Work out each multiplication.
x + 4x − 14 = 6
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
x + 4x = 5x.
x + 4x = 5x
Now we have 5x − 14 = 6.
5x − 14 = 6
We want x by itself.
The − 14 is in the way.
To remove − 14, we add 14.
We do it on both sides, so they stay equal.
5x − 14 + 14 = 6 + 14
− 14 and + 14 make zero.
Cross them out.
6 plus 14 is 20.
6 + 14 = 20
Now we have 5x = 20.
5x = 20
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5x5 = 205
5 divided by 5 is 1.
So just x is left.
20 divided by 5 is 4.
205 = 4
So x = 4.
x = 4
4 Find y
Now we put x = 4 into ③.
y = −2(4) + 7
That is −8 + 7.
y = −8 + 7
So y = −1.
y = −1
So x = 4 and y = −1, both at the same time.
We write it as (4, −1).
On a graph, it is the point where the two lines meet.
(x, y) = (4, −1)
5 Check
Let's check in both equations.
We put x = 4 and y = −1 into ①.
① 2(4) + (−1) = 8 − 1 = 7
We get 7, the same as the right side.
Now ②.
② 4 − 2(−1) = 4 + 2 = 6
We get 6, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get y alone in ①.
Then: put ③ into ②.
Then: find x.
Then: find y.
Last: check.
▶ Watch it solved
The question
Question: solve x − 2y = −2 and 3x + 2y = 34 by substitution.
x − 2y = −2①
We call them ① and ②.
We want the x and y that make both true.
3x + 2y = 34②
1 Get x alone in ①
No letter is alone yet.
In ①, x has just 1 in front.
So we get x alone there first.
We want x by itself.
The − 2y is in the way.
To remove − 2y, we add 2y.
We do it on both sides, so they stay equal.
x − 2y + 2y = −2 + 2y
− 2y and + 2y make zero.
Cross them out.
Write what is left.
x = −2 + 2y
We write the y part first.
Each part keeps its sign.
x is by itself.
We call it ③.
x = 2y − 2③
2 Put ③ into ②
In ③, x is now alone.
x is the same as 2y − 2.
So in ②, we write (2y − 2) in place of x.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3(2y − 2) + 2y = 34
3 Find y
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 × 2y + 3 × (−2) + 2y = 34
Careful!
Not only the first part!
−2 is multiplied by 3 too.
Careful: not 6y − 2 ✗
Work out each multiplication.
6y − 6 + 2y = 34
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the y terms go together.
6y + 2y = 8y.
6y + 2y = 8y
Now we have 8y − 6 = 34.
8y − 6 = 34
We want y by itself.
The − 6 is in the way.
To remove − 6, we add 6.
We do it on both sides, so they stay equal.
8y − 6 + 6 = 34 + 6
− 6 and + 6 make zero.
Cross them out.
34 plus 6 is 40.
34 + 6 = 40
Now we have 8y = 40.
8y = 40
8y means 8 times y.
We want just y.
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8y8 = 408
8 divided by 8 is 1.
So just y is left.
40 divided by 8 is 5.
408 = 5
So y = 5.
y = 5
4 Find x
Now we put y = 5 into ③.
x = 2(5) − 2
That is 10 − 2.
x = 10 − 2
So x = 8.
x = 8
So x = 8 and y = 5, both at the same time.
We write it as (8, 5).
On a graph, it is the point where the two lines meet.
(x, y) = (8, 5)
5 Check
Let's check in both equations.
We put x = 8 and y = 5 into ①.
① 8 − 2(5) = 8 − 10 = −2
We get −2, the same as the right side.
Now ②.
② 3(8) + 2(5) = 24 + 10 = 34
We get 34, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get x alone in ①.
Then: put ③ into ②.
Then: find y.
Then: find x.
Last: check.
▶ Watch it solved
The question
Question: solve a + 2b = 7 and 3a − b = 7 by substitution.
a + 2b = 7①
We call them ① and ②.
We want the a and b that make both true.
3a − b = 7②
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.
a + 2b − 2b = 7 − 2b
+ 2b and − 2b make zero.
Cross them out.
Write what is left.
a = 7 − 2b
We write the b part first.
Each part keeps its sign.
a is by itself.
We call it ③.
a = −2b + 7③
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(−2b + 7) − b = 7
3 Find b
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 × (−2b) + 3 × 7 − b = 7
Careful!
Not only the first part!
7 is multiplied by 3 too.
Careful: not −6b + 7 ✗
Work out each multiplication.
−6b + 21 − b = 7
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.
−6b − b = −7b
Now we have −7b + 21 = 7.
−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.
−7b + 21 − 21 = 7 − 21
+ 21 and − 21 make zero.
Cross them out.
7 minus 21 is −14.
7 − 21 = −14
Now we have −7b = −14.
−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.
−7b−7 = −14−7
−7 divided by −7 is 1.
So just b is left.
−14 divided by −7 is 2.
−14−7 = 2
So b = 2.
b = 2
4 Find a
Now we put b = 2 into ③.
a = −2(2) + 7
That is −4 + 7.
a = −4 + 7
So a = 3.
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.
(a, b) = (3, 2)
5 Check
Let's check in both equations.
We put a = 3 and b = 2 into ①.
① 3 + 2(2) = 3 + 4 = 7
We get 7, the same as the right side.
Now ②.
② 3(3) − 2 = 9 − 2 = 7
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.
▶ Watch it solved
