Forming Expressions, Equations, and Formulae

An expression is a statement made in the language of mathematics. We ascribe meaning to the symbols we use so they may be used instead of words, and so an expression must also work as a sentence.

$4x^2-32$

means “four times ‘x’ squared minus thirty-two”. If an expression doesn’t make sense as a sentence, then it probably doesn’t make sense as an expression.

Here is some nonsense 

$!x^( times +$

This in not an expression, because it doesn’t make any sense. As a sentence this would be “factorial times ‘x’ to the power of open bracket, times plus”. It does not follow the syntax of maths. 

We may form expressions from the processes we know a variable goes through, for example, if we know that the variable $x$ is added to $16$, and then all of that is multiplied by $2.5$, we can form an expression for this, which is 

$2.5(x+16)$

The order of operations is important, remember BIDMAS, the reason we put $x+16$ in the brackets is because we want that to be done first.

An equation is when two equal expressions are joined by an equals sign. They can have at most one unknown value. This means that an equation may be solvable. For example

$14x +15 = 71$

may be solved, my manipulating the equation:

$14x = 56$
$x=4$

The equation has been solved and the unknown value, $x$ has been found. 

You may have to formulate equations, it is very similar to formulating expressions. If we say that $6$ plus twelve lots of $x$ is equal to $66$, then we can formulate the equations

$6 +12x = 66$

Once again, take care of the order of operations.

A formula gives the relationship between variables. A simple example might be the circumference of a circle

$C=pi times d$

Where $C$ is the circumference and $d$ is the diameter of the circle.

A formula is distinct from an equation, as an equation can have at most one variable. So instead of showing a relation between variables, an equation can be used to calculate the value of one variable. Whereas a formula will need the other variables for the unknown variable to be found. 

Pythagoras’ theorem is the formula $a^2+b^2=c^2$. If the length of the base and the length of the height of a right angled triangle is known, then they can be substituted in to $a$ and $b$ to find the unknown variable, the length of the hypotenuse, $c$.

Formulae are created to help with making the same calculations with a change in parameters. 

It is expected of us to be able to create formula given some information. Firstly we must find out what the subject of the formula is, and then what that value is dependent on. 

For example, if a car rental company charges $£50$ per day to rent a car, plus an additional $10text{p}$ a mile, then we can create a formula for the cost of the car hire.

The variables the cost depends on are the number of days the car is hired (was shall call this $t$ for time), and the number of miles the car travels (we shall call this $d$ for distance). We shall then call the cost of the car hire $C$,

$C=50t+0.1d$

We multiply the number of days by $50$ and the distance in miles by $0.1$ as $10text{p}$ is $£0.10$, this gives us $C$ in pounds.