Before solving an equation you may need to form it from the information given in the question.

- An
**expression**is an algebraic statement**without**an equals sign e.g. or - Sometimes we need to form expressions to help us express unknown values
- If a value is unknown you can represent it by a letter such as
- You can turn common phrases into expressions
- Here you can represent the "something" by any letter

2 less than "something" Double the amount of "something" 5 lots of "something" 3 more than "something" Half the amount of "something"

- Here you can represent the "something" by any letter
- You might need to use brackets to show the correct order
- "something" add 1 then multiplied by 3
- which simplifies to

- "something" multiplied by 3 then add 1
- which simplifies to

- "something" add 1 then multiplied by 3
- To make the expression as easy as possible choose the smallest value to be represented by a letter
- If Adam is 10 years
**younger**than Barry, then Barry is 10 years**older**than Adam- Represent Adam's age as then Barry's age is
- This makes the algebra easier, rather than calling Barry's age and Adam's age

- If Adam's age is half of Barry's age then...
- Barry's age is double Adam's age
- So if Adam's age is then Barry's age is
- This makes the algebra easier, rather than using for Barry's age and for Adams's age

- If Adam is 10 years

- An
**equation**is simply an expression with an**equals**sign that can then be solved - You will first need to form an expression and make it equal to a value or another expression
- It is useful to know alternative words for basic operations:
- For
**addition**: sum, total, more than, increase, etc - For
**subtraction**: difference, less than, decrease, etc - For
**multiplication**: product, lots of, times as many, double, triple etc - For
**division**: shared, split, grouped, halved, quartered etc

- For
- Using the first example above
- If Adam is 10 years younger than Barry and the sum of their ages is 25, you can find out how old each one is
- Represent Adam's age as then Barry's age is
- We can solve the equation or

- If Adam is 10 years younger than Barry and the sum of their ages is 25, you can find out how old each one is
- Sometimes you might have
**two**unrelated unknown values (*x*and*y*) and have to use the given information to form two**simultaneous**equations

At a theatre the price of a child's ticket is and the price of an adult's ticket is .

Write equations to represent the following statements:

a)

An adult's ticket is double the price of a child's ticket.

b)

A child's ticket is £7 cheaper than an adult's ticket.

c)

The total cost of 3 children's tickets and 2 adults' tickets is £45.

a)

Adult = 2 × Child

equivalently you could put

b)

Rewrite as:

Adult = Child + £7

equivalently you could put or

c)

Total means add

3 × Child + 2 × Adult = £45

Many questions involve having to form and solve equations from information given about things relating to shapes, like lengths or angles.

- Read the question carefully to decide if it involves area, perimeter or angles
- If no diagram is given it is almost always a good idea to quickly sketch one
- Add any information given in the question to the diagram
- This information will normally involve expressions in terms of one or two variables

- If the question involves
**perimeter**, figure out which sides are equal length and add these together- Consider the properties of the given shape to decide which sides will have equal lengths
- In a square or rhombus, all four sides are equal
- In a rectangle or parallelogram, opposite sides are equal
- If a triangle is given, are any of the sides equal length?

- Consider the properties of the given shape to decide which sides will have equal lengths
- If the question involves
**area**, write down the necessary**formula**for the area of that shape- If it is an uncommon shape you may need to
**split it up**into two or more common shapes that you can work out areas for - In the case you will have to split the length and width up accordingly

- If it is an uncommon shape you may need to
- Remember that a
**regular**polygon means all the**sides are equal**length- For example, a regular pentagon with side length 2
*x*– 1 has 5 equal sides so its perimeter is 5(2*x*– 1)

- For example, a regular pentagon with side length 2
- If one of the shapes is a circle or part of a circle, use π throughout rather than multiplying by it and ending up with long decimals

- If no diagram is given it is almost always a good idea to quickly sketch one
- Add any information given in the question to the diagram
- This information will normally involve expressions in terms of one or two variables
- Consider the properties of angles within the given shape to decide which sides will have equal lengths
- If a triangle is given, how many of the angles are equal?
- An isosceles triangle has two equal angles
- An equilateral triangle has three equal angles
- Consider
**angles in parallel lines**(alternative, corresponding, co-interior) - In a
**parallelogram**or**rhombus**, opposite angles are equal and all four sum to 360° - A
**kite**has one equal pair of opposite angles - If the question involves angles, use the formula for the
**sum of the interior angles**of a polygon - For a polygon of
*n*sides, the sum of the angles will be 180°×(*n*- 2) - Remember that a regular polygon means all the angles are equal
- If a question involves an
**irregular**polygon, assume all the**angles are****different**unless told otherwise - Look out for key information that can give more information about the angles
- For example, a trapezium "with a line of symmetry" will have two pairs of equal angles

- Read the question carefully to decide if it involves surface area or volume
- Mixing these up is a common mistake made in GCSE exams
- If no diagram is given it is almost always a good idea to quickly sketch one
- Add any information given in the question to the diagram
- This information will normally involve expressions in terms of one or two variables
- Consider the properties of the given shape to decide which sides will have equal lengths
- In a cube all sides are equal
- All
**prisms**have the same shape (**cross section**) at the front and back - Pyramids normally have 1/3 in the formula
- If the question involves volume, write down the necessary
**formula**for the area of that shape - If it is an uncommon shape the exam question will give you the formula that you need
- Substitute the expressions for the side lengths into the formula
- Remember to include brackets around any expression that you substitute in
- It the question involves surface area,
- STEP 1

Write down the number of faces the shape has and if any are the same - STEP 2

Identify the 2D shape of each face and write down the formula for the area of each one - STEP 3

Substitute the given expressions into the formula for each one, being careful to identify the correct expression for the dimension - You may need to add or subtract some expressions
- STEP 4

Add the expressions together, double checking that you have one for each of the faces - Remember to consider any faces that may be hidden in the diagram

- Use pencil to annotate the diagrams carefully
- You may find that most of your working for a question is on the diagram itself
- Read the question carefully - don't find the area if it wants the perimeter, don't find the volume if it wants the surface area, etc!

A rectangle has a length of cm and a width of cm.

Its perimeter is equal to 22 cm.

a)

Use the above information to form an equation in terms of *x*.

The perimeter of a rectangle is 2(length) × 2(width).

P = 2(3*x* + 1) + 2(2*x* – 5)

Expand the brackets.

2(3*x* + 1) + 2(2*x* – 5) = 6*x *+ 2 + 4*x* - 10

Simplify.

6*x *+ 2 + 4*x* – 10 = 10*x* – 8

Set equal to the value given for the perimeter.

10*x* – 8 = 22

This equation can be simplified.

**5 x – 4 = 11**

b)

Solve the equation from part (a) to find the value of *x*.

Add 4 to both sides.

5*x* – 4 = 11

5*x* = 15

Divide both sides by 5.

*x *= 3

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