Forming & Solving Equations (CIE IGCSE Maths: Extended)

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

How do I form an expression?

• 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"

• 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 
• 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

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
• 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 
• Sometimes you might have two unrelated unknown values (x and y) and have to use the given information to form two simultaneous equations

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

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, 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?
• 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
• Remember that a regular polygon means all the sides are equal length
  • For example, 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, use π throughout rather than multiplying by it and ending up with long decimals

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

• 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  

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
• 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