# Expressions and Equations

## Expressions and Equations Revision

**Expressions and Equations**

**Expressions** and **equations** use numbers to represent words and letters, making complicated problems easier to understand and solve.

**Expressions**: algebraic statements which may consist of numbers, letters, and operations, but do not contain equals signs. For example x+1

**Equations**: uses an equals sign to equate an equation to something. For example x+1=4

**Substitution**

**Substitution** means replacing letters or symbols in algebraic **equations**/**expressions** with numbers.

For example, in the **equation**

a = 2b + 5

We can use **substitution** to replace the letters to use this equation. We could find the value of a when b=2 by using **substitution**, for example:

a=2\times2 +5

We need to remember **BIDMAS** when computing this (division/multiplication come before addition/subtraction):

a = 4+5\\

a=9

We can also use **substitution** to replace words in **equations** too (these are often formulae):

\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}

We could work out speed when distance = 3\text{ km} and time = 0.5\text{ hours},

\text{Speed}=\dfrac{3}{0.5}

\text{Speed}=6\text{km/h}

**Substitution – Harder**

Sometimes, we may have to use **substitution** in more difficult **equations** or **expressions**, but it works in the same way.

For example,** equations** may have quite a few variables:

y = 2a + a + 4b -12c + 16c -2d + 4

You may notice there are multiples of some of the variables, for example, it contains 2a and a. Therefore, we will need to collect like terms before **substituting**:

2a + a = 3a

-12c+16c=4c

So, the simplified **equation** is,

y = 3a+4b+4c-2d+4

Now we can use **substitution**:

With more complicated** equations** like this, it is even more important to remember BIDMAS. It is often helpful to use brackets to make it easier to **substitute** into complicated **equations**.

For example, work out y given,

a=2\\ b=5\\ c=3\\ d=1\\

Remember that a(b)=a\times b,

y = 3(2)+4(5)+4(3)-2(1)+4

y=6+20+12-2+4

y=40

**Forming Expressions**

When forming an expression, we will often use an algebraic variable (usually a letter like x, y, a, b, but can be any letter) to **represent an unknown**.

- For example, Sophie has some sweets, and she buys 3 more sweets at the shop. Represent the number of sweets Sophie has using an algebraic
**expression**:

We do not know how many sweets Sophie started with, so we can use a variable to represent this. As we are not told which variable to use, we can choose any letter, let’s choose s.

Sophie has s sweets and buys 3 more, so overall Sophie has:

s + 3 sweets.

**Expressions** can be more complicated, where there is **more than one unknown**.

- Sophie, Chloe, and Ella have some sweets. Represent the total number of sweets as an
**expression**.

Here, we have 3 unknowns, so we need 3 algebraic symbols. We will call the number of sweets Sophie has s, the number of sweets Chloe has c, and for Ella e:

s + c + e

**Forming Equations**

Similarly, when forming **equations** we often use an algebraic variable (usually a letter like x, y, a, b, but can be any letter) to **represent an unknown**.

- For example, Sophie has some sweets, and she loses 3 of them. Sophie now has y sweets. Represent y in an
**equation**:

We do not know how many sweets Sophie started with, so we can use s.

Sophie has s sweets and loses 3, resulting in her having y sweets. We can represent this as:

s - 3 = y sweets.

**Forming Equations – Harder**

**Equations** can be more complicated, for example:

- Sophie and Ella each have some lollipops. Their friend Chloe has 2 times the amount of lollipops Sophie has. In total, they have 30 lollipops. Represent this as an equation.

We can call the number of sweets Sophie has s and the number Ella has e. If Chloe has double the amount of sweets Sophie has, we can call this 2s:

s + 2s + e = 30

As we have 2 separate lots of s here, we can group the s‘s together:

3s+e=30

**Example 1: Expressions**

Barney has 3 siblings. His youngest sister Joy is half of his age, his brother Jason is 6 years older than Barney, and his older sister Daisy is 4 years older than Jason. By representing Joy’s age as x, write expressions to represent the age of each siblings in terms of x.

**[4 marks]**

Joy: x

Barney is double the age of Joy: 2x

Jason is 6 years older than Barney: 2x+6

Daisy is 4 years older than Jason: 2x+6+4 = 2x+10

**Example 2: Equations**

The total length of all of the sides of this 2D shape is 100. Represent this as an equation.

**[4 marks]**

To find the total length, we add all the the sides together:

2x+x+3+5x+6y-5+y+1=100

We can collect the x‘s, the y‘s and the numbers to simplify this equation:

8x+7y-1=100

We can now add 1 to each side:

8x+7y=101

## Expressions and Equations Example Questions

**Question 1: **Tesni is buying food for a picnic. She buys 6 pizzas for £x each, 12 packets of crisps for £y each, and 10 sausage rolls for £z each. Write an expression for the total amount Tesni spends.

**[3 marks]**

6 pizzas for £x each: 6x

12 packets of crisps for £y each: 12y

10 sausage rolls for £z each: 10z

Total: 6x + 12y+10z

**Question 2: **Theo, Harry, Emerson and William are hosting a party. 35 people are attending. Theo invited x people and Harry invited y people. Emerson invited double the amount of people Theo invited, and William invited 5 less people than Theo.

Represent the number of people at the party with an equation in terms of x.

**[3 marks]**

Theo: x

Harry: y

Emerson: 2x

William:2x-5

Total:

x+y+2x+2x-5=35\\

y+5x-5=35\\

y+5x=40

**Question 3:**

K=6l + 3m -\dfrac{n}{3}\\

l = 2\\

m = 4\\

n = 12

Work out the value of K

**[3 marks]**

Using substitution,

K=6(2)+3(4)-\dfrac{12}{3}\\ K=12+12-4\\ K=20