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# Solving Equations

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## Solving Equations

To rearrange formula and make one of the variables the **subject**, you use the same rules as you use to solve linear equations. The only difference is you are collecting the variable you want on one side and everything else on the other.

**Try and follow these steps:**

**1.** Get rid of any square roots by squaring both sides.

**2.** Get rid of any brackets by multiplying out.

**3. **Get rid of any fractions by multiplying **all terms** by the denominator.

**4. **Rearrange by collecting the letter you want on one side and everything else on the other. Remember that every term has its own sign in front of it. Negative terms need to be added to both sides to get rid of them.

**5. **Factorise the side your variable is on so there is only one of it.

**6. **Divide both sides by whatever the variable is multiplied by.

**Here's an example:**

**Here is an example rearranging: a(a ^{2} + y) = b^{3} making y the subject.**

**Simultaneous equations** are two equations both with two different variables in them.

You cannot solve them on their own (as they have infinite solutions!) so we have to merge them to end up with a linear equation which we can solve (see earlier section on 'Linear equations').

**You can follow some simple steps which we will show you using this example:**

**Solve:**

y = 2x - 3 |

5x - 2y = 8 |

Simultaneous equations will only have **one** pair of values for x and y that work in both equations. Here's how to find them:

**1.** First write both equations in the form: ax + by = c

-2x + y = -3 |

5x - 2y = 8 |

**2.** Now, you need to match up the numbers in front of **either** the x's or the y's. To do this you **multiply** one (or both) of the equations to get a match. If you multiply a whole equation by something you must remember to multiply every term. We are going to multiply the first equation by 2 as this will match the y's.

-4x + 2y = -6 |

5x - 2y = 8 |

**3.** Now you can **merge** the two equations by adding them or subtracting them from each other (whichever gets rid of the y's!). In this case we will add them to get rid of the y's (careful with negative signs!).

**x = 2**

**(-4x + 5x just gives x, 2y + -2y disappears and -6 + 8 gives 2)**

**4.** Sometimes you will be left with a Linear Equation in x which you have to solve but, in this case we were just left with the answer. Now **substitute** your value of x into either of the original equations to find y. We will use the first one of the original equations:

**y = 2x - 3**

**y = 4 - 3, which gives y = 1**

**5.** Now substitute your answers into the other equation to check that they work!

**5x - 2y = 8**

Putting x = 2 and y = 1 into this gives 10 - 2 = 8 which works!!!

So our answer is x = 2, y = 1

The method we've just shown you is called **elimination.**

To see how to solve simultaneous equations using a graph see the graph section.

If you are asked to solve a problem by forming an equation it's worth remembering that equations are just shorthand and you should try to write them in words first before using your highly tuned mathematical skill to write them in symbols.

**Here's an example:**

The perimeter of a rectangle is 98cm. One side is 7cm longer than the other. Form an equation and hence find the length of each side.

**If we call the short side x we can draw the following diagram:**

**Now, in words...**

All the sides added together gives 98.

**So as an equation...**

x + (x + 7) + x + (x + 7) = 98

Simplify this equation and solve it!

**4x + 14 = 98**

**4x = 84**

**x = 21**

So the shorter side is 21cm and the longer side is 28cm.

Just do a quick check to make sure that 21 + 28 + 21 + 28 = 98

And it does. Hurrah!

This is dealt with in **Approximations**.