GCSE Maths: Substituting negative numbers (Higher)

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Work in the order BIDMAS, one step at a time, taking care with the signs.

1. Write down the calculation. Copy the calculation, and look at each sign carefully.
   5 + (−3)
2. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   5 + (−3) = 5 − 3 = 2 (adding a negative number is the same as subtracting)
   = 2

Check. Put the numbers straight into a calculator: 5 + (−3) gives 2. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Work in the order BIDMAS, one step at a time, taking care with the signs.

1. Write down the calculation. Copy the calculation, and look at each sign carefully.
   −4 − (−6)
2. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −4 − (−6) = −4 + 6 = 2 (subtracting a negative number is the same as adding)
   = 2

Check. Put the numbers straight into a calculator: −4 − (−6) gives 2. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Work in the order BIDMAS, one step at a time, taking care with the signs.

1. Write down the calculation. Copy the calculation, and look at each sign carefully.
   3 × (−4)
2. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   3 × (−4) = −12 (a negative times a positive gives a negative)
   = −12

Check. Put the numbers straight into a calculator: 3 × (−4) gives −12. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Work in the order BIDMAS, one step at a time, taking care with the signs.

1. Write down the calculation. Copy the calculation, and look at each sign carefully.
   −2 × (−7)
2. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   −2 × (−7) = 14 (a negative times a negative gives a positive)
   = 14

Check. Put the numbers straight into a calculator: −2 × (−7) gives 14. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Work in the order BIDMAS, one step at a time, taking care with the signs.

1. Write down the calculation. Copy the calculation, and look at each sign carefully.
   12 ÷ (−3)
2. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   12 ÷ (−3) = −4 (a negative divided by a positive gives a negative)
   = −4

Check. Put the numbers straight into a calculator: 12 ÷ (−3) gives −4. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Work in the order BIDMAS, one step at a time, taking care with the signs.

1. Write down the calculation. Copy the calculation, and look at each sign carefully.
   −20 ÷ (−4)
2. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   −20 ÷ (−4) = 5 (a negative divided by a negative gives a positive)
   = 5

Check. Put the numbers straight into a calculator: −20 ÷ (−4) gives 5. ✓

Watch out. Work in the order of BIDMAS: multiply and divide before you add and subtract.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x + 5
2. Substitute: x = −2. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x + 5 = −2 + 5
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −2 + 5 = 3
   = 3

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −2 + 5 gives 3. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   3x
2. Substitute: x = −2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   3x = 3 × (−2)
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   3 × (−2) = −6 (a negative times a positive gives a negative)
   = −6

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 3 × (−2) gives −6. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   4 − x
2. Substitute: x = −2. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   4 − x = 4 − (−2)
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   4 − (−2) = 4 + 2 = 6 (subtracting a negative number is the same as adding)
   = 6

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 4 − (−2) gives 6. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x − 1
2. Substitute: x = −2. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x − 1 = −2 − 1
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −2 − 1 = −3
   = −3

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −2 − 1 gives −3. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   2x + 7
2. Substitute: x = −2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   2x + 7 = 2 × (−2) + 7
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   2 × (−2) = −4 (a negative times a positive gives a negative)
   = −4 + 7
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −4 + 7 = 3
   = 3

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 2 × (−2) + 7 gives 3. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   10 + 3x
2. Substitute: x = −2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   10 + 3x = 10 + 3 × (−2)
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   3 × (−2) = −6 (a negative times a positive gives a negative)
   = 10 + (−6)
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   10 + (−6) = 10 − 6 = 4 (adding a negative number is the same as subtracting)
   = 4

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 10 + 3 × (−2) gives 4. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   a + b
2. Substitute: a = −3, b = 4. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   a + b = −3 + 4
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −3 + 4 = 1
   = 1

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −3 + 4 gives 1. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   ab
2. Substitute: a = −3, b = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   ab = −3 × 4
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   −3 × 4 = −12 (a negative times a positive gives a negative)
   = −12

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −3 × 4 gives −12. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   a − b
2. Substitute: a = −3, b = 4. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   a − b = −3 − 4
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −3 − 4 = −7
   = −7

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −3 − 4 gives −7. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   2a + 3b
2. Substitute: a = −3, b = 4. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   2a + 3b = 2 × (−3) + 3 × 4
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   2 × (−3) = −6 (a negative times a positive gives a negative)
   3 × 4 = 12
   = −6 + 12
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −6 + 12 = 6
   = 6

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 2 × (−3) + 3 × 4 gives 6. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   b − a
2. Substitute: b = 4, a = −3. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   b − a = 4 − (−3)
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   4 − (−3) = 4 + 3 = 7 (subtracting a negative number is the same as adding)
   = 7

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 4 − (−3) gives 7. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x²
2. Substitute: x = −5. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x² = (−5)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−5)² = (−5) × (−5) = 25 (a negative times a negative is positive)
   = 25

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−5)² gives 25. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   −x²
2. Substitute: x = −5. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   −x² = −(−5)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−5)² = (−5) × (−5) = 25 (a negative times a negative is positive)
   = −25

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −(−5)² gives −25. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   (−x)²
2. Substitute: x = −5. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   (−x)² = (−(−5))²
3. Deal with the double minus sign. A minus sign in front of a negative number changes it to a positive: −(−4) = 4.
   −(−5) = +5 = 5 (two minus signs next to each other make a plus)
   = 5²
4. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   5² = 5 × 5 = 25
   = 25

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−(−5))² gives 25. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x² + 3
2. Substitute: x = −5. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x² + 3 = (−5)² + 3
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−5)² = (−5) × (−5) = 25 (a negative times a negative is positive)
   = 25 + 3
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   25 + 3 = 28
   = 28

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−5)² + 3 gives 28. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   3x²
2. Substitute: x = −5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   3x² = 3 × (−5)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−5)² = (−5) × (−5) = 25 (a negative times a negative is positive)
   = 3 × 25
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   3 × 25 = 75
   = 75

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 3 × (−5)² gives 75. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   F = 9C5 + 32
2. Substitute: C = −10. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   F = 9 × (−10)5 + 32
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   9 × (−10) = −90 (a negative times a positive gives a negative)
   F = −905 + 32
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   −905 means −90 ÷ 5 = −18 (a negative divided by a positive gives a negative)
   F = −18 + 32
5. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −18 + 32 = 14
   F = 14
6. Write the answer with its units. The answer should say what was being worked out.
   F = 14°F

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (9 × (−10))/5 + 32 gives 14. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   F = 9C5 + 32
2. Substitute: C = −5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   F = 9 × (−5)5 + 32
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   9 × (−5) = −45 (a negative times a positive gives a negative)
   F = −455 + 32
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   −455 means −45 ÷ 5 = −9 (a negative divided by a positive gives a negative)
   F = −9 + 32
5. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −9 + 32 = 23
   F = 23
6. Write the answer with its units. The answer should say what was being worked out.
   F = 23°F

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (9 × (−5))/5 + 32 gives 23. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   F = 9C5 + 32
2. Substitute: C = −40. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   F = 9 × (−40)5 + 32
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   9 × (−40) = −360 (a negative times a positive gives a negative)
   F = −3605 + 32
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   −3605 means −360 ÷ 5 = −72 (a negative divided by a positive gives a negative)
   F = −72 + 32
5. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −72 + 32 = −40
   F = −40
6. Write the answer with its units. The answer should say what was being worked out.
   F = −40°F

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (9 × (−40))/5 + 32 gives −40. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   ab
2. Substitute: a = −2, b = −3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   ab = −2 × (−3)
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   −2 × (−3) = 6 (a negative times a negative gives a positive)
   = 6

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −2 × (−3) gives 6. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   a²b
2. Substitute: a = −2, b = −3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   a²b = (−2)² × (−3)
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−2)² = (−2) × (−2) = 4 (a negative times a negative is positive)
   = 4 × (−3)
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   4 × (−3) = −12 (a negative times a positive gives a negative)
   = −12

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−2)² × (−3) gives −12. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   a² + b²
2. Substitute: a = −2, b = −3. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   a² + b² = (−2)² + (−3)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−2)² = (−2) × (−2) = 4 (a negative times a negative is positive)
   (−3)² = (−3) × (−3) = 9 (a negative times a negative is positive)
   = 4 + 9
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   4 + 9 = 13
   = 13

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−2)² + (−3)² gives 13. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   a − b
2. Substitute: a = −2, b = −3. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   a − b = −2 − (−3)
3. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   −2 − (−3) = −2 + 3 = 1 (subtracting a negative number is the same as adding)
   = 1

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −2 − (−3) gives 1. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   ab²
2. Substitute: a = −2, b = −3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   ab² = −2 × (−3)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−3)² = (−3) × (−3) = 9 (a negative times a negative is positive)
   = −2 × 9
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   −2 × 9 = −18 (a negative times a positive gives a negative)
   = −18

Check. Put the numbers straight into a calculator, with brackets round negative numbers: −2 × (−3)² gives −18. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   (a + b)²
2. Substitute: a = −2, b = −3. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   (a + b)² = (−2 + (−3))²
3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   −2 + (−3) = −2 − 3 = −5 (adding a negative number is the same as subtracting)
   = (−5)²
4. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−5)² = (−5) × (−5) = 25 (a negative times a negative is positive)
   = 25

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−2 + (−3))² gives 25. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   B = 40 − 25t
2. Substitute: t = 1. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   B = 40 − 25 × 1
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   25 × 1 = 25
   B = 40 − 25
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   40 − 25 = 15
   B = 15
5. Write the answer. The answer should say what was being worked out.
   B = £15

Check. Put the numbers straight into a calculator: 40 − 25 × 1 gives 15. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   B = 40 − 25t
2. Substitute: t = 2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   B = 40 − 25 × 2
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   25 × 2 = 50
   B = 40 − 50
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   40 − 50 = −10
   B = −10
5. Write the answer. The answer should say what was being worked out.
   B = −£10

Check. Put the numbers straight into a calculator: 40 − 25 × 2 gives −10. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Write the formula, replace each letter by its value, then work out the right-hand side using the order BIDMAS.

1. Write down the formula. This is the rule we are going to use.
   B = 40 − 25t
2. Substitute: t = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign.
   B = 40 − 25 × 3
3. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   25 × 3 = 75
   B = 40 − 75
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   40 − 75 = −35
   B = −35
5. Write the answer. The answer should say what was being worked out.
   B = −£35

Check. Put the numbers straight into a calculator: 40 − 25 × 3 gives −35. ✓

Watch out. A number next to a letter means multiply: 3x is 3 × x. If x = 5, then 3x = 15, not 35.

Method. Interpret the negative number in the context.

1. Look at the sign. A negative balance means she has spent more than she had.
   B = −£35
2. Say what it means. She owes the bank money.
   Jo is £35 overdrawn.

Check. She had £40 and spent 3 × 25 = £75. £40 − £75 = −£35. ✓

Watch out. A negative answer can make sense in real life: overdrafts, temperatures below zero, and so on.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x + y2
2. Substitute: x = −4, y = 6. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x + y2 = −4 + 62
3. Top of the fraction first: do the additions and subtractions. Inside the top of the fraction we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   −4 + 6 = 2
   = 22
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   22 means 2 ÷ 2 = 1
   = 1

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−4 + 6)/2 gives 1. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   yx
2. Substitute: y = 6, x = −4. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   yx = 6−4
3. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   6−4 means 6 ÷ −4 = −1.5 (a negative divided by a positive gives a negative)
   = −1.5

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 6/(−4) gives −1.5. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   3xy
2. Substitute: x = −4, y = 6. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   3xy = 3 × (−4)6
3. Top of the fraction first: do the multiplications and divisions. Inside the top of the fraction we still use BIDMAS, so multiplying and dividing come before adding and subtracting.
   3 × (−4) = −12 (a negative times a positive gives a negative)
   = −126
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   −126 means −12 ÷ 6 = −2 (a negative divided by a positive gives a negative)
   = −2

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (3 × (−4))/6 gives −2. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x²y + 2
2. Substitute: x = −4, y = 6. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x²y + 2 = (−4)²6 + 2
3. Top of the fraction first: work out the powers (indices). Inside the top of the fraction we still use BIDMAS, so powers come first.
   (−4)² = (−4) × (−4) = 16 (a negative times a negative is positive)
   6 + 2 = 8
   = 168
4. Divide the top by the bottom. The fraction bar means divide. Both the top and the bottom are now single numbers, so we can divide.
   168 means 16 ÷ 8 = 2
   = 2

Check. Put the numbers straight into a calculator, with brackets round negative numbers: ((−4)²)/(6 + 2) gives 2. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   (x − 3)²
2. Substitute: x = −1. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   (x − 3)² = (−1 − 3)²
3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   −1 − 3 = −4
   = (−4)²
4. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−4)² = (−4) × (−4) = 16 (a negative times a negative is positive)
   = 16

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−1 − 3)² gives 16. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   3(2 − x)
2. Substitute: x = −1. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   3(2 − x) = 3 × (2 − (−1))
3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   2 − (−1) = 2 + 1 = 3 (subtracting a negative number is the same as adding)
   = 3 × 3
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   3 × 3 = 9
   = 9

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 3 × (2 − (−1)) gives 9. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   (x + 4)(x − 2)
2. Substitute: x = −1. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   (x + 4)(x − 2) = (−1 + 4) × (−1 − 2)
3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   −1 + 4 = 3
   −1 − 2 = −3
   = 3 × (−3)
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   3 × (−3) = −9 (a negative times a positive gives a negative)
   = −9

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−1 + 4) × (−1 − 2) gives −9. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   5 − (x + 2)
2. Substitute: x = −1. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   5 − (x + 2) = 5 − (−1 + 2)
3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   −1 + 2 = 1
   = 5 − 1
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   5 − 1 = 4
   = 4

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 5 − (−1 + 2) gives 4. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   2x² − 3xy + y²
2. Substitute: x = −2, y = 3. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   2x² − 3xy + y² = 2 × (−2)² − 3 × (−2) × 3 + 3²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−2)² = (−2) × (−2) = 4 (a negative times a negative is positive)
   3² = 3 × 3 = 9
   = 2 × 4 − 3 × (−2) × 3 + 9
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   2 × 4 = 8
   3 × (−2) = −6 (a negative times a positive gives a negative)
   = 8 − (−6) × 3 + 9
5. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   −6 × 3 = −18 (a negative times a positive gives a negative)
   = 8 − (−18) + 9
6. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   8 − (−18) = 8 + 18 = 26 (subtracting a negative number is the same as adding)
   = 26 + 9
7. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   26 + 9 = 35
   = 35

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 2 × (−2)² − 3 × (−2) × 3 + 3² gives 35. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   2x² − 3xy + y²
2. Substitute: x = 3, y = −2. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   2x² − 3xy + y² = 2 × 3² − 3 × 3 × (−2) + (−2)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   3² = 3 × 3 = 9
   (−2)² = (−2) × (−2) = 4 (a negative times a negative is positive)
   = 2 × 9 − 3 × 3 × (−2) + 4
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   2 × 9 = 18
   3 × 3 = 9
   = 18 − 9 × (−2) + 4
5. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   9 × (−2) = −18 (a negative times a positive gives a negative)
   = 18 − (−18) + 4
6. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   18 − (−18) = 18 + 18 = 36 (subtracting a negative number is the same as adding)
   = 36 + 4
7. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   36 + 4 = 40
   = 40

Check. Put the numbers straight into a calculator, with brackets round negative numbers: 2 × 3² − 3 × 3 × (−2) + (−2)² gives 40. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   x² − y²
2. Substitute: x = −3, y = −5. Substitute means replace each letter with its number. Put brackets round any negative number so that its sign is safe.
   x² − y² = (−3)² − (−5)²
3. Work out the powers (indices). BIDMAS tells us the order: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction. A power such as 5² means 5 × 5, and it is done before multiplying, dividing, adding or subtracting.
   (−3)² = (−3) × (−3) = 9 (a negative times a negative is positive)
   (−5)² = (−5) × (−5) = 25 (a negative times a negative is positive)
   = 9 − 25
4. Do the additions and subtractions. Addition and subtraction come last in BIDMAS. If there is more than one, work from left to right.
   9 − 25 = −16
   = −16

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−3)² − (−5)² gives −16. ✓

Watch out. Put a negative number in brackets before you square it: (−3)² = 9, but −3² = −9. The brackets decide what gets squared.

Key words. To substitute means to replace each letter with the number it stands for. An expression is a mixture of letters, numbers and operations (like 3x + 2). A formula is a rule that links quantities and has an equals sign (like A = lw). BIDMAS is the order for working out: Brackets, Indices, Division/Multiplication, Addition/Subtraction.

Method. Replace each letter by its value, then work out the answer using the order BIDMAS.

1. Write down the expression. Start by copying the expression exactly.
   (x − y)(x + y)
2. Substitute: x = −3, y = −5. Substitute means replace each letter with its number. A number written next to a letter means multiply, so we show the × sign. Put brackets round any negative number so that its sign is safe.
   (x − y)(x + y) = (−3 − (−5)) × (−3 + (−5))
3. Brackets first: do the additions and subtractions. Inside the brackets we still use BIDMAS. Only adding and subtracting are left, so work from left to right.
   −3 − (−5) = −3 + 5 = 2 (subtracting a negative number is the same as adding)
   −3 + (−5) = −3 − 5 = −8 (adding a negative number is the same as subtracting)
   = 2 × (−8)
4. Do the multiplications and divisions. In BIDMAS, multiplication and division come before addition and subtraction. If there is more than one, work from left to right.
   2 × (−8) = −16 (a negative times a positive gives a negative)
   = −16

Check. Put the numbers straight into a calculator, with brackets round negative numbers: (−3 − (−5)) × (−3 + (−5)) gives −16. ✓

Watch out. Always put brackets round a negative number when you substitute it. Without them the minus sign is easy to lose, for example 5 − −3 looks wrong but 5 − (−3) is clear.

Method. Compare the two answers, then use what you know about expanding.

1. Compare. Both parts gave the same value.
   −16 and −16
2. Explain. Expanding (x − y)(x + y) gives x² + xy − xy − y², and the middle terms cancel.
   (x − y)(x + y) = x² − y². This is the difference of two squares (Chapter 7).

Check. It works for other numbers too, for example x = 4, y = 1: 16 − 1 = 15 and (3)(5) = 15. ✓

Watch out. Equal answers for one pair of numbers suggest an identity; expanding proves it.