**A group of people went to London Zoo, where they bought a number of adult tickets and childrens' tickets. **

**Mr. and Mrs. Brown took their daughter Libby and paid £68, whereas Mr. and Mrs. White took their three sons, Tom, Dick and Harry and paid £100. **

**The final family group attending were the Blacks. **

**Can you work out how much Ms. Black, with her children Jack and Maisy, would have to pay?**

Problems like this are common in maths books and exams.

There are a number of ways you can solve them, such as trial and improvement, but here we will look at a method where we turn the information into two equations and solve them at the same time.

This method is known as **solving simultaneous equations**.

In order to find out how much Ms. Black and her two children will pay, we first need to know the cost of an adult ticket and the cost of a child's ticket.

These are known as variables and we use a letter to stand for each one.

In many simultaneous equations, **x** and **y** are used, but we can use any letter we choose.

Here we will use **a** (cost of an adult) and **c** (cost of a child).

So, Mr. and Mrs. Brown and their daughter becomes 2 adults and one child paying £68 or **2a + c = 68**.

Mr. and Mrs. White and their 3 sons becomes 2 adults and 3 children paying £100 or **2a + 3c = 100**.

We now have our two equations, and we need to find values for **a** and **c** which work in both equations **at the same time (simultaneously)**.

If we line up the equations, one above the other with the one with the biggest values on top it will help:

**2a + 3c = 100**

**2a + c = 68**

We can see that both equations have **2a** in them, but the top one has **2c** more than the bottom one and the cost is £32 more, so **2c = 32**.

If 2 children cost £32 then one must cost **£16**, so **c =16**.

Knowing these two facts, we can know work out a.

If **2c + c = 68** and **c = 16** then **2c + 16 = 68**.

We can solve this equation easily by subtracting 16 from both sides giving **2a = 52**, and dividing by 2 gives **a = 26**.

So an adult ticket costs **£26**.

Now we have our two ticket costs, it is a good idea to check that they work for the other family (the equation we have not yet used).

We had **2a + 3c = 100 **and substituting **a = 26** and **c= 16**** **gives **2 x 26 + 3 x 16 = 52 + 48 = 100**, which is correct.

Now we can find the cost for the Black family, which is one adult and 2 children:

**26 + 2 x 16 = 58**, so they would pay **£58**.

Now, let's look at another pair of simultaneous equations.

This time just the equations have been given without the real-life problem.

**Solve 5x + y = 17 and 3x + y = 11**

Let's number our equations (1) and (2) with the bigger values on top then subtract equation (2) from (1):

**5x + y = 17 ****(1)**

**3x + y = 11 ****(2)**

**2x = 6 ****(1) - (2)**

**x = 3 ** (divide by 2)

Substitute **x = 3** into (2) and solve the equation [you could use equation (1) if you prefer]:

**3 x 3 + y = 11**

**9 + y = 11**

**y = 2**

**So x = 3 and y = 2**

Let's check these values work with equation (1) [i.e. the equation we did __ not__ use to find our values]

**5 x 3 + 2 = 17** (which is correct)

Let's look at one more problem to check you have it.

**Solve 5x - y = 17 and 2x + y = 11**

**5x - y = 17 ** **(1)**

**2x + y = 11** ** (2)**

Notice that the y's have the same coefficients (number in front) and so this is the variable which will be eliminated when we combine the equations.

Because the **signs in front of the equations are different **(one positive and one negative) this time we need to **add** them:

**7x = 28 ****(1) + (2)**

**x = 4 ****[divide by 7]**

**2 x 4 + y = 11 ****[substitute into (2)]**

**8 + y = 11**

**y = 3**

**Let's check this in (1):**

**5 x 4 - 3 = 17 [****correct]**

**So x = 4 and y = 3.**

**TOP TIPS:**

Remember the variable with **equal coefficients** will be **eliminated** when the equations are combined.

If the **signs** in front are the **same**, you **subtract**, but if they are** different** then you **add**.

Remember the catchphrase: **Same Sign Subtract (SSS) **to help you commit this rule to memory.

In this activity, we will find the values of variable by solving pairs of simultaneous equations.

You may want to have a pen and paper handy to record your working and so you can compare it to our maths teacher's method.