Forming Equations | Edexcel GCSE Maths: Foundation Revision Notes 2017

Forming & Solving Equations

Why do I need to form expressions and equations?

Sometimes a question gives you information about a situation in words
- You need to be able to form expressions and equations from this information
- You can then solve the equations that you have formed

How do I form an expression?

An expression is an algebraic statement without an equals sign, e.g. $3 x + 7$ or $2 (x^{2} - 14)$
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 $x$
You can turn common phrases into expressions
- Here you can represent the "something" by any letter
  
  2 less than "something" $x - 2$
  
  Double the amount of "something" $2 x$
  
  5 lots of "something" $5 x$
  
  3 more than "something" $x + 3$
  
  Half the amount of "something" $\frac{1}{2} x or \frac{x}{2}$
You might need to use brackets to show the correct order
- "something" add 1 then multiplied by 3
  - $(x + 1) \times 3$ which simplifies to $3 (x + 1)$
- "something" multiplied by 3 then add 1
  - $(x \times 3) + 1$ which simplifies to $3 x + 1$
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 $x$ then Barry's age is $x + 10$
  - This makes the algebra easier, rather than calling Barry's age $x$ and Adam's age $x - 10$
- 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 $x$ then Barry's age is $2 x$
  - This makes the algebra easier, rather than using $x$ for Barry's age and $\frac{1}{2} x$ for Adams's age

How do I form an equation?

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 example, Adam is 10 years younger than Barry and the sum of their ages is 26
- You can find out how old each one is
  - Represent Adam's age as $x$ then Barry's age is $x + 10$
- Then form the equation by adding together the ages and making the expression equal to 26
  - $x + x + 10 = 26$ or $2 x + 10 = 26$
- This is now an equation that can be solved to find the value of $x$
Sometimes you may have two unrelated unknown values (x and y) and have to use the given information to form two simultaneous equations

Worked example

At a theatre the price of a child's ticket is $£ x$ and the price of an adult's ticket is $£ y$ .

Write equations to represent the following statements:

(a)

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

Adult = 2 × Child

$y = 2 x$

x = \frac{1}{2} y

is also correct

(b)

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

Rewrite as:

Adult = Child + £7

$y = x + 7$

$x = y - 7$ or $y - x = 7$ are also correct

(c)

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

Total means add

3 × Child + 2 × Adult = £45

$3 x + 2 y = 45$

Forming Equations from Shapes

How do I form an equation involving the area or perimeter of a 2D shape?

Read the question carefully to decide if it involves area or perimeter
- It is almost always a good idea to quickly sketch a diagram if one is not given
- 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
- 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
  - A triangle may have 0, 2 or 3 equal sides
  - A kite has two pairs of equal adjacent sides
If the question involves area
- Write down the necessary formula for the area of that shape
  - You may need to split it up into two or more common shapes that you can work out areas for
  - In this case you will have to split the length and width up accordingly
Remember that a regular polygon means all the sides are equal length
- E.g. A regular pentagon with side length 2x – 1 has 5 equal sides so its perimeter is 5(2x – 1)
If one of the shapes is a circle or part of a circle, keep working in terms of
- Avoid working with long decimals

How do I form an equation involving angles in a 2D shape?

Read the question carefully to decide if it involves angles
- It is almost always a good idea to quickly sketch a diagram if one is not given
- 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
- A square or a rectangle has four angles of 90°
- In a parallelogram or rhombus, opposite angles are equal and all four sum to 360°
- A triangle may have 0, 2 or 3 equal angles
- A kite has one equal pair of opposite angles
- Consider angles in parallel lines (alternative, corresponding, co-interior)
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 interior 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 words that can give more information about the angles
- E.g. A trapezium "with a line of symmetry" will have two pairs of equal angles

EPS Notes fig4

How do I form an equation involving the surface area or volume of a 3D shape?

Read the question carefully to decide if it involves surface area or volume
- It is almost always a good idea to quickly sketch a diagram if one is not given
- 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
If the question involves volume, write down the necessary formula for the volume 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
If the question involves surface area
- You may need to add or subtract some expressions
- Remember to consider any faces that may be hidden in the diagram
- To calculate the surface area
  - Write down the number of faces the shape has and if any are the same
  - Identify the 2D shape of each face and write down the formula for the area of each one
  - Substitute the given expressions into the formula for each one
  - Add the expressions together, double checking that you have one for each of the faces

Examiner Tip

Most given diagrams are not drawn to scale.
- Do not measure lengths and angles from these diagrams.
Use pencil to annotate the diagrams carefully.
- Pencil means that you can easily erase mistakes.
- You may find that most of your working for a question is on the diagram itself
Read the question carefully!
- Make a final check that the quantity you have calculated (perimeter, area etc.) is what the question asked for.
- Mixing up quantities, particularly surface area and volume, is a common mistake at GCSE.

Worked example

A rectangle has a length of $3 x + 1$ cm and a width of $2 x - 5$ cm.

Its perimeter is equal to 22 cm.

(a)

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

Sketch a diagram and label it

Rectangle with length 3x+1 cm and width 2x-5 cm

The perimeter of a rectangle is 2(length) + 2(width)

P = 2(3x + 1) + 2(2x – 5)

Expand the brackets

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

Simplify

6x + 2 + 4x – 10 = 10x – 8

Set equal to the value given for the perimeter

10x – 8 = 22

This equation can be simplified by dividing all parts by 2

5x – 4 = 11

(b)

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

Add 4 to both sides

5x – 4 = 11

5x = 15

Divide both sides by 5

x = 3

Forming Equations (Edexcel GCSE Maths: Foundation)

Revision Note

Forming & Solving Equations

Why do I need to form expressions and equations?

How do I form an expression?

How do I form an equation?

Worked example

Forming Equations from Shapes

How do I form an equation involving the area or perimeter of a 2D shape?

How do I form an equation involving angles in a 2D shape?

How do I form an equation involving the surface area or volume of a 3D shape?

Examiner Tip

Worked example

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Author: Naomi C

2 less than "something"	$x - 2$
Double the amount of "something"	$2 x$
5 lots of "something"	$5 x$
3 more than "something"	$x + 3$
Half the amount of "something"	$\frac{1}{2} x or \frac{x}{2}$