# Equation with Two Variables: How Many Answers? (2)

In this worksheet, students must find how many integer solutions there are to a given equation with two variables.

Key stage:  KS 2

Curriculum topic:   Maths and Numerical Reasoning

Curriculum subtopic:   Equations and Algebra

Difficulty level:

### QUESTION 1 of 10

In this worksheet, all the answers must be whole positive numbers or zero.

An equation with two variables involves two numbers which must be found.

"I think of two numbers.

When I double the first and add this to the second, I get 8."

Written using algebra, where a and b stand for my two numbers, we get:

2a + b = 8

This means that twice a added to b must come to 8.

a and b are whole positive numbers or zero.

What could they be?

If a is 0, b must be 8.

If a is 1, b must be 8 - 2 = 6.

If a is 2, b must be 8 - 4 = 4.

If a is 3, b must be 8 - 6 = 2.

If a is 4, b must be 8 - 8 = 0.

The results can be shown in a table:

a   b
0 8
1 6
2 4
3 2
4 0

There are 5 solutions.

"I think of two numbers.

When I double one and add the other, I get 3."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 3

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 1
a = 1
b = 3

"I think of two numbers.

When I double one and add the other, I get 3."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 3

How many solutions does it have where a and b are whole numbers or zero?

"I think of two numbers.

When I double one and add the other, I get 4."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 4

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 0
a = 1
b = 2
a = 2
b = 4

"I think of two numbers.

When I double one and add the other, I get 4."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 4

How many solutions does it have where a and b are whole numbers or zero?

"I think of two numbers.

When I double one and add the other, I get 5."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 5

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 5
a = 1
b = 3
a = 2
b = 1

"I think of two numbers.

When I double one and add the other, I get 5."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 5

How many solutions does it have where a and b are whole numbers or zero?

"I think of two numbers.

When I double one and add the other, I get 7."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 7

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 7
a = 1
b = 1
a = 2
b = 5
a = 3
b = 3

"I think of two numbers.

When I double one and add the other, I get 7."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 7

How many solutions does it have where a and b are whole numbers or zero?

"I think of two numbers.

When I double one and add the other, I get 9."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 9

How many solutions does it have where a and b are whole numbers or zero?

"I think of two numbers.

When I double one and add the other, I get 10."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 10

How many solutions does it have where a and b are whole numbers or zero?

• Question 1

"I think of two numbers.

When I double one and add the other, I get 3."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 3

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 3
a = 1
b = 1
EDDIE SAYS
2 × 0 + 3 = 0 + 3 = 3
2 × 1 + 1 = 2 + 1 = 3
• Question 2

"I think of two numbers.

When I double one and add the other, I get 3."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 3

How many solutions does it have where a and b are whole numbers or zero?

2
EDDIE SAYS
2 × 0 + 3 = 0 + 3 = 3
2 × 1 + 1 = 2 + 1 = 3
• Question 3

"I think of two numbers.

When I double one and add the other, I get 4."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 4

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 4
a = 1
b = 2
a = 2
b = 0
EDDIE SAYS
2 × 0 + 4 = 0 + 4 = 4
2 × 1 + 2 = 2 + 2 = 4
2 × 2 + 0 = 4 + 0 = 4
• Question 4

"I think of two numbers.

When I double one and add the other, I get 4."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 4

How many solutions does it have where a and b are whole numbers or zero?

3
EDDIE SAYS
2 × 0 + 4 = 0 + 4 = 4
2 × 1 + 2 = 2 + 2 = 4
2 × 2 + 0 = 4 + 0 = 4
• Question 5

"I think of two numbers.

When I double one and add the other, I get 5."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 5

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 5
a = 1
b = 3
a = 2
b = 1
EDDIE SAYS
2 × 0 + 5 = 0 + 5 = 5
2 × 1 + 3 = 2 + 3 = 5
2 × 2 + 1 = 4 + 1 = 5
• Question 6

"I think of two numbers.

When I double one and add the other, I get 5."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 5

How many solutions does it have where a and b are whole numbers or zero?

3
EDDIE SAYS
2 × 0 + 5 = 0 + 5 = 5
2 × 1 + 3 = 2 + 3 = 5
2 × 2 + 1 = 4 + 1 = 5
• Question 7

"I think of two numbers.

When I double one and add the other, I get 7."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 7

Match the answers for b to the answers for a, to give the correct solutions.

## Column B

a = 0
b = 7
a = 1
b = 5
a = 2
b = 3
a = 3
b = 1
EDDIE SAYS
2 × 0 + 7 = 0 + 7 = 7
2 × 1 + 5 = 2 + 5 = 7
2 × 2 + 3 = 4 + 3 = 7
2 × 3 + 1 = 6 + 1 = 7
• Question 8

"I think of two numbers.

When I double one and add the other, I get 7."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 7

How many solutions does it have where a and b are whole numbers or zero?

4
EDDIE SAYS
2 × 0 + 7 = 0 + 7 = 7
2 × 1 + 5 = 2 + 5 = 7
2 × 2 + 3 = 4 + 3 = 7
2 × 3 + 1 = 6 + 1 = 7
• Question 9

"I think of two numbers.

When I double one and add the other, I get 9."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 9

How many solutions does it have where a and b are whole numbers or zero?

5
EDDIE SAYS
2 × 0 + 9 = 0 + 9 = 9
2 × 1 + 7 = 2 + 7 = 9
2 × 2 + 5 = 4 + 5 = 9
2 × 3 + 3 = 6 + 3 = 9
2 × 4 + 1 = 8 + 1 = 9
• Question 10

"I think of two numbers.

When I double one and add the other, I get 10."

Here it is using algebra, where a and b stand for my two numbers.

2a + b = 10

How many solutions does it have where a and b are whole numbers or zero?

6
EDDIE SAYS
2 × 0 + 10 = 0 + 10 = 10
2 × 1 + 8 = 2 + 8 = 10
2 × 2 + 6 = 4 + 6 = 10
2 × 3 + 4 = 6 + 4 = 10
2 × 4 + 2 = 8 + 2 = 10
2 × 5 + 0 = 10 + 0 = 10
---- OR ----

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