Basic Algebra

Algebra follows the same rules you have used with numbers, but because the number is unknown it can look very different.

Variables:

If we do not know the value of a number, we can call it \( x \), \( a \) or whichever letter we like.

Expressions

\[3 \times 5 = 15 \]

If we multiply two numbers, such as 3 and 5, we can write the answer.

\[3 \times x \]

We cannot do this with algebra because we do not know what the number is.

\[3 \times x = 3x \]

The \(\times \) and the \( x \) look very similar! Thankfully we never have to show the (\times \) sign in algebra.

\[3 \times x + 10 = 3x + 10 \]

If we added 10 to \( 3x \) we can write this as \( 3x + 10\)

Substitution

If we find out what number the letter represents, we can find the value of an expression by using substitution.

\[ 3x + 10\]

If \( x = 4 \) we can find the value of \( 3x + 10 \) by replacing the x for 4. But be careful! We must first put back in the times sign.

\[ 3 \times x + 10\]

\[ 3 \times 4 + 10\]

\[ 12 + 10\]

\[ 22\]

Equations

An equation is where we have two expressions equal to each other. You can spot an equation because it will have an equals sign.

\[ 3x + 5 = 14\]

In this equation, x is a value that can be found.

Formula

A formula is a statement linking more than one variable (can be real-world).

\[ A = \frac12 \times b \times h\]

You might recognise this formula as the area of a triangle. Each letter means something. \(b\) is the base of the triangle and \(h\) is the height.

Identity

An identity is a statement that is true for all values of the variable.

\[ 3x = x + x + x\]

This identity is true for all values of \(x\).

\[3 \times 5 = 5 + 5 + 5\]

\[3 \times 10 = 10 + 10 + 10\]

\[3 \times 999 = 999 + 999 + 999\]

We can show identities with three equals signs (but they will probably use two in your exam).

\[ 3x \equiv x + x + x\]