Substitution

What you'll learn

How to replace letters in an expression or formula with given numbers.
How to spot hidden multiplication, such as $3 f$ , $b c$ and $m (n - 2)$ .
How to deal with negative numbers, powers and fractions safely.
How to show enough working to earn method marks.

1. What substitution means

In algebra, letters usually stand for numbers. Substitution means putting the given number in place of the letter, then calculating the answer.

Definition

Key words

A variable is a letter that represents a number, such as $x$ , $f$ or $t$ .
An expression is a mathematical phrase, such as $3 f + 2 g$ .
A formula is an equation that links variables, such as $v = u + a t$ .
To evaluate an expression means to find its numerical value.

Key Idea

Replace first, calculate second

Substitution is a two-stage process: first replace each letter with its given number, then use the normal order of operations to calculate.

Example

Substituting into a simple expression

Suppose $f = 6$ and $g = 4$ . Find the value of $3 f + 2 g$ .

Replace $f$ with 6 and $g$ with 4. Remember that $3 f$ means $\times f$ .
$3 f + 2 g = 3 (6) + 2 (4)$
Multiply before adding.
$3 (6) + 2 (4) = 18 + 8$
Add the two parts.
$18 + 8 = 26$
The value of the expression is 26.

2. Hidden multiplication and BIDMAS

In algebra, multiplication signs are often left out.

For example:

$4 d$ means $\times d$ .
$j k$ means $\times k$ .
$m (n - 2)$ means $\times (n - 2)$ .

You must still follow BIDMAS, the order of operations:

Brackets
Indices, meaning powers such as $x^2$
Division and Multiplication
Addition and Subtraction

Common Mistake

Do not join the numbers

If $d = 7$ , then $4 d$ means $\times 7$ , not 47. The missing operation is multiplication.

Example

Substituting into a formula

A formula is $c = 5 d - 8$ . Find $c$ when $d = 7$ .

Replace $d$ with 7.
$c = 5 (7) - 8$
Multiply first.
$c = 35 - 8$
Subtract.
$c = 27$
So the value of $c$ is 27.

3. Substituting negative numbers

Negative numbers need extra care. The safest habit is to put the negative number in brackets when you substitute it.

For example, if $r = - 3$ , write $4 (- 3)$ rather than $4 - 3$ . This keeps the multiplication clear.

Tip

Bracket negative substitutions

When a variable has a negative value, put it in brackets straight away. This avoids most sign errors.

Example

One positive and one negative value

A formula is $Q = 6 p + 4 r$ . Find $Q$ when $p = 5$ and $r = - 3$ .

Substitute $p = 5$ and $r = - 3$ .
$Q = 6 (5) + 4 (- 3)$
Do the multiplications.
$Q = 30 + (- 12)$
Adding a negative is the same as subtracting.
$Q = 30 - 12$
Calculate the final value.
$Q = 18$

Example

Subtracting a negative value

Find $A$ when $A = p - 2 q$ , $p = - 3$ and $q = - 8$ .

Substitute both values using brackets.
$A = (- 3) - 2 (- 8)$
Work out the multiplication part first.
$A = (- 3) - (- 16)$
Subtracting a negative becomes adding.
$A = - 3 + 16$
Finish the calculation.
$A = 13$

4. Brackets in formulas

Some formulas contain brackets already. Brackets tell you to calculate that part first.

If the formula is $L = m (n - 4)$ , this means:

first work out $n - 4$ ;
then multiply by $m$ .

Example

A formula with brackets

Find $L$ when $L = m (n - 4)$ , $m = 8$ and $n = 9$ .

Substitute the values into the formula.
$L = 8 (9 - 4)$
Work out the bracket.
$L = 8 (5)$
Multiply.
$L = 40$

Sometimes letters are next to each other, such as $b c$ . This also means multiplication.

Example

Multiplying two negative values

Find $a$ when $a = 6 b c$ , $b = - 2$ and $c = - 5$ .

Substitute both negative values using brackets.
$a = 6 (- 2) (- 5)$
Multiply the two negative numbers first.
$(- 2) (- 5) = 10$
Multiply by 6.
$a = 6 (10)$
The value of $a$ is 60.

5. Powers and squared numbers

An index is a power. For example, $x^2$ means $\times x$ , so it is read as “ $x$ squared”.

When substituting into a square, substitute the value first, then square it.

Common Mistake

Squaring a negative number

If $x = - 3$ , then $x^2 = (-3)^2 = 9$ . The brackets matter because the whole value of $x$ is being squared.

Example

A formula with a square

Find $w$ when $w = 4x^2 + 7$ and $x = - 3$ .

Substitute $x = - 3$ using brackets.
$w = 4(-3)^2 + 7$
Square the negative number first.
$w = 4 (9) + 7$
Multiply before adding.
$w = 36 + 7$
Finish the calculation.
$w = 43$

Some expressions mix powers, multiplication and subtraction.

Example

A mixed expression with several letters

Find the value of $b^2 - 4ac$ when $a = 2$ , $b = - 5$ and $c = 3$ .

Substitute the values carefully.
$b^2 - 4ac = (-5)^2 - 4(2)(3)$
Square first.
$5)^2 = 25$
Work out the multiplication part.
$4 (2) (3) = 24$
Subtract.
$25 - 24 = 1$

6. Fractions in formulas

Some formulas include fractions. The line in a fraction means “divide by”.

Definition

Parts of a fraction

The numerator is the top part of a fraction.
The denominator is the bottom part of a fraction.

For formulas like $\frac{1}{2}bh$ , multiplying by $12\frac{1}{2}$ means finding half of the product $bh$ .

Example

A formula with one half

Find $A$ when $\frac{1}{2}bh$ , $b = 10$ and $h = 7$ .

Substitute the values.
$\frac{1}{2}(10)(7)$
Multiply the base and height.
$\frac{1}{2}(70)$
Find half of 70.
$A = 35$

For a formula with a larger fraction, work out the numerator and denominator separately before dividing.

Example

A formula with a fraction bar

Find $s$ when $\frac{v^2 - u^2}{2a}$ , $v = 8$ , $u = 4$ and $a = 2$ .

Substitute all the values using brackets where helpful.
$\frac{8^2 - 4^2}{2(2)}$
Work out the powers in the numerator.
$\frac{64 - 16}{2(2)}$
Finish the numerator and denominator.
$\frac{48}{4}$
Divide.
$s = 12$

7. Longer formulas: stay organised

Longer formulas are not harder if you keep the working tidy. Substitute first, then deal with each part using BIDMAS.

A common example is $\frac{1}{2}at^2$ . This has multiplication, a fraction and a square, so order matters.

Example

A longer substitution

Find $s$ when $\frac{1}{2}at^2$ , $u = - 4$ , $a = 6$ and $t = 3$ .

Substitute the values.
$\frac{1}{2}(6)(3)^2$
Work out the square first.
$\frac{1}{2}(6)(9)$
Work out the multiplication parts.
$s = - 12 + 27$
Add the two terms.
$s = 15$

Exam technique

In the exam

Write the substituted expression before calculating, because this can earn method marks even if the final arithmetic goes wrong.
Put brackets around negative values as soon as you substitute them.
Use BIDMAS carefully: brackets, powers, multiplication or division, then addition or subtraction.
Check whether your answer should be positive or negative, especially when the question includes negative numbers.

Self review

Check yourself

When you see $7 p$ , what operation is hidden between 7 and $p$ ?
Why is $4)^2$ positive when substituting into a formula?
In a formula with a fraction bar, which parts should you calculate before dividing?

Substitution

What you'll learn

How to replace letters in an expression or formula with given numbers.
How to spot hidden multiplication, such as $3 f$ , $b c$ and $m (n - 2)$ .
How to deal with negative numbers, powers and fractions safely.
How to show enough working to earn method marks.

1. What substitution means

In algebra, letters usually stand for numbers. Substitution means putting the given number in place of the letter, then calculating the answer.

Definition

Key words

A variable is a letter that represents a number, such as $x$ , $f$ or $t$ .
An expression is a mathematical phrase, such as $3 f + 2 g$ .
A formula is an equation that links variables, such as $v = u + a t$ .
To evaluate an expression means to find its numerical value.

Key Idea

Replace first, calculate second

Substitution is a two-stage process: first replace each letter with its given number, then use the normal order of operations to calculate.

Example

Substituting into a simple expression

Suppose $f = 6$ and $g = 4$ . Find the value of $3 f + 2 g$ .

Replace $f$ with 6 and $g$ with 4. Remember that $3 f$ means $\times f$ .
$3 f + 2 g = 3 (6) + 2 (4)$
Multiply before adding.
$3 (6) + 2 (4) = 18 + 8$
Add the two parts.
$18 + 8 = 26$
The value of the expression is 26.

2. Hidden multiplication and BIDMAS

In algebra, multiplication signs are often left out.

For example:

$4 d$ means $\times d$ .
$j k$ means $\times k$ .
$m (n - 2)$ means $\times (n - 2)$ .

You must still follow BIDMAS, the order of operations:

Brackets
Indices, meaning powers such as $x^2$
Division and Multiplication
Addition and Subtraction

Common Mistake

Do not join the numbers

If $d = 7$ , then $4 d$ means $\times 7$ , not 47. The missing operation is multiplication.

Example

Substituting into a formula

A formula is $c = 5 d - 8$ . Find $c$ when $d = 7$ .

Replace $d$ with 7.
$c = 5 (7) - 8$
Multiply first.
$c = 35 - 8$
Subtract.
$c = 27$
So the value of $c$ is 27.

3. Substituting negative numbers

Negative numbers need extra care. The safest habit is to put the negative number in brackets when you substitute it.

For example, if $r = - 3$ , write $4 (- 3)$ rather than $4 - 3$ . This keeps the multiplication clear.

Tip

Bracket negative substitutions

When a variable has a negative value, put it in brackets straight away. This avoids most sign errors.

Example

One positive and one negative value

A formula is $Q = 6 p + 4 r$ . Find $Q$ when $p = 5$ and $r = - 3$ .

Substitute $p = 5$ and $r = - 3$ .
$Q = 6 (5) + 4 (- 3)$
Do the multiplications.
$Q = 30 + (- 12)$
Adding a negative is the same as subtracting.
$Q = 30 - 12$
Calculate the final value.
$Q = 18$

Example

Subtracting a negative value

Find $A$ when $A = p - 2 q$ , $p = - 3$ and $q = - 8$ .

Substitute both values using brackets.
$A = (- 3) - 2 (- 8)$
Work out the multiplication part first.
$A = (- 3) - (- 16)$
Subtracting a negative becomes adding.
$A = - 3 + 16$
Finish the calculation.
$A = 13$

4. Brackets in formulas

Some formulas contain brackets already. Brackets tell you to calculate that part first.

If the formula is $L = m (n - 4)$ , this means:

first work out $n - 4$ ;
then multiply by $m$ .

Example

A formula with brackets

Find $L$ when $L = m (n - 4)$ , $m = 8$ and $n = 9$ .

Substitute the values into the formula.
$L = 8 (9 - 4)$
Work out the bracket.
$L = 8 (5)$
Multiply.
$L = 40$

Sometimes letters are next to each other, such as $b c$ . This also means multiplication.

Example

Multiplying two negative values

Find $a$ when $a = 6 b c$ , $b = - 2$ and $c = - 5$ .

Substitute both negative values using brackets.
$a = 6 (- 2) (- 5)$
Multiply the two negative numbers first.
$(- 2) (- 5) = 10$
Multiply by 6.
$a = 6 (10)$
The value of $a$ is 60.

5. Powers and squared numbers

An index is a power. For example, $x^2$ means $\times x$ , so it is read as “ $x$ squared”.

When substituting into a square, substitute the value first, then square it.

Common Mistake

Squaring a negative number

If $x = - 3$ , then $x^2 = (-3)^2 = 9$ . The brackets matter because the whole value of $x$ is being squared.

Example

A formula with a square

Find $w$ when $w = 4x^2 + 7$ and $x = - 3$ .

Substitute $x = - 3$ using brackets.
$w = 4(-3)^2 + 7$
Square the negative number first.
$w = 4 (9) + 7$
Multiply before adding.
$w = 36 + 7$
Finish the calculation.
$w = 43$

Some expressions mix powers, multiplication and subtraction.

Example

A mixed expression with several letters

Find the value of $b^2 - 4ac$ when $a = 2$ , $b = - 5$ and $c = 3$ .

Substitute the values carefully.
$b^2 - 4ac = (-5)^2 - 4(2)(3)$
Square first.
$5)^2 = 25$
Work out the multiplication part.
$4 (2) (3) = 24$
Subtract.
$25 - 24 = 1$

6. Fractions in formulas

Some formulas include fractions. The line in a fraction means “divide by”.

Definition

Parts of a fraction

The numerator is the top part of a fraction.
The denominator is the bottom part of a fraction.

For formulas like $\frac{1}{2}bh$ , multiplying by $12\frac{1}{2}$ means finding half of the product $bh$ .

Example

A formula with one half

Find $A$ when $\frac{1}{2}bh$ , $b = 10$ and $h = 7$ .

Substitute the values.
$\frac{1}{2}(10)(7)$
Multiply the base and height.
$\frac{1}{2}(70)$
Find half of 70.
$A = 35$

For a formula with a larger fraction, work out the numerator and denominator separately before dividing.

Example

A formula with a fraction bar

Find $s$ when $\frac{v^2 - u^2}{2a}$ , $v = 8$ , $u = 4$ and $a = 2$ .

Substitute all the values using brackets where helpful.
$\frac{8^2 - 4^2}{2(2)}$
Work out the powers in the numerator.
$\frac{64 - 16}{2(2)}$
Finish the numerator and denominator.
$\frac{48}{4}$
Divide.
$s = 12$

7. Longer formulas: stay organised

Longer formulas are not harder if you keep the working tidy. Substitute first, then deal with each part using BIDMAS.

A common example is $\frac{1}{2}at^2$ . This has multiplication, a fraction and a square, so order matters.

Example

A longer substitution

Find $s$ when $\frac{1}{2}at^2$ , $u = - 4$ , $a = 6$ and $t = 3$ .

Substitute the values.
$\frac{1}{2}(6)(3)^2$
Work out the square first.
$\frac{1}{2}(6)(9)$
Work out the multiplication parts.
$s = - 12 + 27$
Add the two terms.
$s = 15$

Exam technique

In the exam

Write the substituted expression before calculating, because this can earn method marks even if the final arithmetic goes wrong.
Put brackets around negative values as soon as you substitute them.
Use BIDMAS carefully: brackets, powers, multiplication or division, then addition or subtraction.
Check whether your answer should be positive or negative, especially when the question includes negative numbers.

Self review

Check yourself

When you see $7 p$ , what operation is hidden between 7 and $p$ ?
Why is $4)^2$ positive when substituting into a formula?
In a formula with a fraction bar, which parts should you calculate before dividing?

Substitution

What you'll learn

1. What substitution means

2. Hidden multiplication and BIDMAS

3. Substituting negative numbers

4. Brackets in formulas

5. Powers and squared numbers

6. Fractions in formulas

7. Longer formulas: stay organised

Recap questions

You've reached the end

Substitution

What you'll learn

1. What substitution means

2. Hidden multiplication and BIDMAS

3. Substituting negative numbers

4. Brackets in formulas

5. Powers and squared numbers

6. Fractions in formulas

7. Longer formulas: stay organised

Recap questions

You've reached the end

Substitution

What you'll learn

1. What substitution means

2. Hidden multiplication and BIDMAS

3. Substituting negative numbers

4. Brackets in formulas

5. Powers and squared numbers

6. Fractions in formulas

7. Longer formulas: stay organised

Recap questions

You've reached the end

Substitution Revision Guide

Substitution

What you'll learn

1. What substitution means

2. Hidden multiplication and BIDMAS

3. Substituting negative numbers

4. Brackets in formulas

5. Powers and squared numbers

6. Fractions in formulas

7. Longer formulas: stay organised

Recap questions

You've reached the end

Substitution Revision Guide