Year 6: Pairs of numbers with two unknowns
Part of the Year 6 algebra chapter. Chapter overview · whole chapter on one page
Pairs of numbers with two unknowns
An equation with two letters, such as a + b = 9, can have many answers. Each answer is a pair: a = 1 and b = 8 is one pair, and a = 5 and b = 4 is another.
To find all the pairs, choose a value for one letter, starting from 1. Work out the other letter. Then choose the next value, and keep going until the other letter would be less than 1. A table keeps the list in order, so that no pair is missed or counted twice.
If the question gives a second clue, such as a is bigger than b, use it to cross out pairs that do not fit.
Key ideas
- An equation with two unknowns has many pairs of answers.
- Make a table: choose one letter in order, and work out the other letter each time.
- Check each pair in the original equation.
- Count the rows of the table to find how many pairs there are.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 9
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 8 1 + 8 = 9 2 7 2 + 7 = 9 3 6 3 + 6 = 9 4 5 4 + 5 = 9 5 4 5 + 4 = 9 6 3 6 + 3 = 9 7 2 7 + 2 = 9 8 1 8 + 1 = 9 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 8 and 1
- Count the pairs. Each row of the table is one pair.8 pairs
- Write the answer. 8
Check. Every row was checked in the equation. The table has 8 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.x + 2y = 10
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 2 4 2 + 2 × 4 = 10 4 3 4 + 2 × 3 = 10 6 2 6 + 2 × 2 = 10 8 1 8 + 2 × 1 = 10 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 8 and 1
- Count the pairs. Each row of the table is one pair.4 pairs
- Write the answer. 4
Check. Every row was checked in the equation. The table has 4 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a × b = 24
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 24 1 × 24 = 24 2 12 2 × 12 = 24 3 8 3 × 8 = 24 4 6 4 × 6 = 24 6 4 6 × 4 = 24 8 3 8 × 3 = 24 12 2 12 × 2 = 24 24 1 24 × 1 = 24 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 24 and 1
- Count the pairs. Each row of the table is one pair.8 pairs
- Write the answer. 8
Check. Every row was checked in the equation. The table has 8 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Watch out
Pairs are written in the order of the letters. (1, 8) and (8, 1) are two different pairs for a + b = 9, because a has a different value in each. Both are counted. Read the question to see if the letters have an order.
Exam tip
Use a table. Write the pairs in order, starting with the smallest value, so that you can see when the list is complete.
Exercise 8E Finding values and pairs
Non-calculatorKey words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 4 into the equation. Now there is only one letter left.a + b = 10
- Take 4 from 10. This is how much is left for the other part.10 - 4 = 6
- The number left is the value of the letter. This finds the other letter.b = 6
- Check the pair in the equation. Both sides must be equal.4 + 6 = 10 ✓
- Write the answer. b = 6
Check. 4 + 6 = 10. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put x = 7 into the equation. Now there is only one letter left.x + y = 15
- Take 7 from 15. This is how much is left for the other part.15 - 7 = 8
- The number left is the value of the letter. This finds the other letter.y = 8
- Check the pair in the equation. Both sides must be equal.7 + 8 = 15 ✓
- Write the answer. y = 8
Check. 7 + 8 = 15. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Add 9 to both numbers to find a. Then check the answer.
- Write the equation with b = 9. Put the known number in place of b.a - 9 = 3
- Add 9 to both sides. This undoes the take away 9.a = 3 + 9 = 12
- Write the answer. 12
Check. 12 - 9 = 3. ✓
Watch out. Add to find the bigger number when you subtract.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 1 into the equation. Now there is only one letter left.2a + b = 12
- Work out 2 × 1, then take it from 12. This is how much is left for the other part.12 - 2 = 10
- The number left is the value of the letter. This finds the other letter.b = 10
- Check the pair in the equation. Both sides must be equal.2 + 10 = 12 ✓
- Write the answer. b = 10
Check. 2 + 10 = 12. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 3 into the equation. Now there is only one letter left.2a + b = 12
- Work out 2 × 3, then take it from 12. This is how much is left for the other part.12 - 6 = 6
- The number left is the value of the letter. This finds the other letter.b = 6
- Check the pair in the equation. Both sides must be equal.6 + 6 = 12 ✓
- Write the answer. b = 6
Check. 6 + 6 = 12. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put in the number you know, then work out the other letter.
- Put a = 5 into the equation. Now there is only one letter left.2a + b = 12
- Work out 2 × 5, then take it from 12. This is how much is left for the other part.12 - 10 = 2
- The number left is the value of the letter. This finds the other letter.b = 2
- Check the pair in the equation. Both sides must be equal.10 + 2 = 12 ✓
- Write the answer. b = 2
Check. 10 + 2 = 12. ✓
Watch out. Only one letter is unknown after you put in the number. Work it out from the equation.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 12
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 11 1 + 11 = 12 2 10 2 + 10 = 12 3 9 3 + 9 = 12 4 8 4 + 8 = 12 5 7 5 + 7 = 12 6 6 6 + 6 = 12 7 5 7 + 5 = 12 8 4 8 + 4 = 12 9 3 9 + 3 = 12 10 2 10 + 2 = 12 11 1 11 + 1 = 12 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 11 and 1
- Count the pairs. Each row of the table is one pair.11 pairs
- Write the answer. 11
Check. Every row was checked in the equation. The table has 11 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.x + y = 20, both odd
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 1 19 1 + 19 = 20, both odd 3 17 3 + 17 = 20, both odd 5 15 5 + 15 = 20, both odd 7 13 7 + 13 = 20, both odd 9 11 9 + 11 = 20, both odd 11 9 11 + 9 = 20, both odd 13 7 13 + 7 = 20, both odd 15 5 15 + 5 = 20, both odd 17 3 17 + 3 = 20, both odd 19 1 19 + 1 = 20, both odd - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 19 and 1
- Count the pairs. Each row of the table is one pair.10 pairs
- Write the answer. 10
Check. Every row was checked in the equation. The table has 10 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a + b = 14, a > b
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 8 6 8 + 6 = 14, and 8 is bigger 9 5 9 + 5 = 14, and 9 is bigger 10 4 10 + 4 = 14, and 10 is bigger 11 3 11 + 3 = 14, and 11 is bigger 12 2 12 + 2 = 14, and 12 is bigger 13 1 13 + 1 = 14, and 13 is bigger - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 13 and 1
- Count the pairs. Each row of the table is one pair.6 pairs
- Write the answer. 6
Check. Every row was checked in the equation. The table has 6 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.x + 3y = 15
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 3 4 3 + 3 × 4 = 15 6 3 6 + 3 × 3 = 15 9 2 9 + 3 × 2 = 15 12 1 12 + 3 × 1 = 15 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 12 and 1
- Count the pairs. Each row of the table is one pair.4 pairs
- Write the answer. 4
Check. Every row was checked in the equation. The table has 4 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Add the two equations to find x. Then use either equation to find y.
- Write the two equations. Both are true for the same x and y.x + y = 9 and x - y = 3
- Add the two equations. The y terms cancel out: +y and -y.2x = 12
- Divide by 2. This gives x.x = 12 ÷ 2 = 6
- Find y. Put x = 6 into x + y = 9.6 + y = 9, so y = 3
- Write the answer. 6
Check. 6 + 3 = 9 and 6 - 3 = 3. ✓
Watch out. Check both equations with both numbers.
| x | y |
|---|---|
| 1 | 7 |
| 3 | 6 |
| 5 | 5 |
| 7 | 4 |
| 9 | ? |
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Use the equation x + 2y = 15 with x = 9.
- Write the equation. The table shows pairs for this equation.x + 2y = 15
- Put x = 9 into the equation. Now there is only y left.9 + 2y = 15
- Take 9 from 15. This leaves 2y.2y = 15 - 9 = 6
- Divide by 2. This gives y.y = 6 ÷ 2 = 3
- Write the answer. 3
Check. 9 + 2 × 3 = 15. ✓
Watch out. Check that the other rows of the table work the same way.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Put the values in the equation and see if both sides are equal.
- Write the equation. We will test the pair.2a + b = 15
- Put in a = 5 and b = 6. Replace each letter by its number.2 × 5 + 6 = 10 + 6
- Compare the answer with 15. It must be exactly 15.10 + 6 = 16, and 16 is not 15
- Write the answer. No
Check. Since 16 ≠ 15, the pair is not a solution. ✓
Watch out. Check the whole equation, not one part of it.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.a × b = 36, a < b
- Start with a = 1 and work out b. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
a b Check 1 36 1 × 36 = 36, and 1 is smaller 2 18 2 × 18 = 36, and 2 is smaller 3 12 3 × 12 = 36, and 3 is smaller 4 9 4 × 9 = 36, and 4 is smaller - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 4 and 9
- Count the pairs. Each row of the table is one pair.4 pairs
- Write the answer. 4
Check. Every row was checked in the equation. The table has 4 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.