Equations and Formulas

Equations

Equations are mathematical expressions that say that two things are equal. Equations always include equals signs \(``="\). For example,

So, an equation is a type of mathematical sentence which includes `equals' as its verb.

The equation above has the solution, \(x = 5\). There's an article about solving equations that will take you through the steps of finding this solution.

Formulas

Formulas are equations satisfying a special property. They not only have to include 'equals', they have to include two or more variables. Variables are symbols like \(x\), \(y\), \(V\) and \(h\) that can take on many different values. They always take the place of numbers.

For example

• \(V = lwh\) is a formula
• \(S = ut + \dfrac{1}{2}at^2\) is a formula
• \(y = 3x + 7\) is a formula
• \(x = 3x - 4\) is an equation, but not a formula as it only includes the one variable, \(x\).

Example 1

Example 2

Mathematicians often develop formulas to model real world situations like this one.

The Subject of a Formula

The subject of a formula is a variable that sits on its own on one side of the formula (usually the left). It is followed by an equals sign, and the other side of the formula (usually the right) includes an expression in the remaining variables that tells you what the subject is equal to.

For example, the subject of Gus the snail's formula:

The subject of the formula:

Changing the Subject of a Formula

One important use of algebra is that it gives us the ability to rearrange a formula to make a different variable its subject. Let's finish off by looking at some examples.

Example 1

Gus the snail is now interested in the number of moths in his garden. He needs to make moths the subject of his formula:

Example 2

Suppose we want to find the required height to produce a cylinder of base radius \(2 \text{ cm}\) and volume \(20 \pi \text{ cm}^3\). We need to make \(h\) the subject of the volume formula:

• Start with \(V = \pi r^2 h\)
• Divide both sides by \(\pi\) to give \(\dfrac{V}{\pi} = r^2 h\).
• Divide both sides by \(r^2\) to give \(\dfrac{V}{\pi r^2} = h\).
• Change the sides over so that \(h\) is on the left: \(h = \dfrac{V}{\pi r^2}\).

Now we can apply our new formula for the height of a cylinder to calculate: