Forming Equations (Edexcel GCSE Maths: Foundation)

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Naomi C

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Naomi C

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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 plus 7 or 2 left parenthesis x squared minus 14 right parenthesis
  • 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 minus 2
      Double the amount of "something" 2 x
      5 lots of "something" 5 x
      3 more than "something" x plus 3
      Half the amount of "something" 1 half x space or space x over 2
  • You might need to use brackets to show the correct order
    • "something" add 1 then multiplied by 3
      • left parenthesis x plus 1 right parenthesis cross times 3 which simplifies to 3 left parenthesis x plus 1 right parenthesis
    • "something" multiplied by 3 then add 1
      • left parenthesis x cross times 3 right parenthesis plus 1 which simplifies to 3 x plus 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 plus 10
      • This makes the algebra easier, rather than calling Barry's age x and Adam's age x minus 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 1 half 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 plus 10
    • Then form the equation by adding together the ages and making the expression equal to 26
      • x plus x plus 10 space equals space 26 or 2 x plus 10 space equals space 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

bold italic y bold equals bold 2 bold italic x

 x equals 1 half y is also correct
 
(b)

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

Rewrite as:

Adult = Child + £7

bold italic y bold equals bold italic x bold plus bold 7

 x equals y minus 7 or y minus x equals 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

bold 3 bold italic x bold plus bold 2 bold italic y bold equals bold 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 pi
    • 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°
    • 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 plus 1 cm and a width of 2 x minus 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

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.