Forming 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

$3 x + 2 y = 45$