In the activity 'Simultaneous Equations - substitution' we learnt how to use the substitution method to solve two** linear** simultaneous equations. If you have not completed that activity yet you should do so before continuing with this one, as in this activity we extend this idea to deal with situations where one equation is linear but the other is not (usually **quadratic**). The method of substitution will be the same but the problems will become more complicated and lengthy. Here is an example to demonstrate how to go about this.

Solve the simultaneous equations

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

**x² + y² = 13 ****(2)**

In this question equation **(1) is linear** and **(2) is quadratic. ** We always substitute the linear into the quadratic equation, but first, we must rearrange **(1)** so that either **x** or **y **is the subject? In this case, because equation **(2)** contains **x² **and** y² **** **it does not matter which so let's make x the subject.

Adding 'y' to both sides gives us

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

Now we can substitute this into **(2)**

**(1 + y)² + y² = 13 **** (2)**

Now, we need to expand the brackets and simplify

**1 + 2y + y² + y² = 13**

**2y² + 2y - 12 = 0**

This is now a **quadratic equation in y** and needs to be solved using any of the methods for solving quadratics. Firstly we can divide the whole equation by 2 to make life easier.

**y² + y - 6 = 0**

This equation will **factorise** into two brackets (always try to factorise before you turn to any other method for solving quadratics, and usually they will factorise).

**(y + 3)(y - 2) = 0**

Either** y + 3 = 0 or y - 2 = 0**

So** y = -3**** or**** y = 2**

So we have two possible values for y and each will have a corresponding value for x. To find the x-values always substitute the y-values into the linear equation.

When** y = -3 x = 1 + -3** so **x = -2**

When** ****y = 2 x = 1 + 2**** so ****x = 3**

So the two pairs of solutions are **x = 3, y = 2** or **x = -2, y = -3. **

These are sometimes written as** co-ordinates** **(3 , 2) **or** (-2 , -3).**

At this point, we could check our solutions by substituting them into equation (2) but to save time we will not do this in this activity.

It will always be the case with these types of simultaneous equation that there will be** two pairs of solutions **and you need to find them both.

Now, it's time to try some yourself.