# Equations and identities

An equation is a statement with an equals sign, stating that two expressions are equal in value, for example $$3x + 5 = 11$$

Solving an equation means finding the value or values for which the two expressions are equal. This means equations are not always true. In the example above, $$3x + 5 = 11$$, the only correct solution for $$x$$ is 2.

An identity is an equation which is always true, no matter what values are substituted. $$2x + 3x = 5x$$ is an identity because $$2x + 3x$$ will always equal $$5x$$ regardless of the value of $$x$$. Identities can be written with the sign ≡, so the example could be written as $$2x + 3x ≡ 5x$$.

### Example

Show that $$x = 2$$ is the solution of the equation $$3x + 5 = 11$$

BIDMAS means the multiplication is carried out before the addition:

$3x + 5 = 3 \times 2 + 5 = 6 + 5 = 11$

Question

Say whether each of the following is an identity or an equation

• $5x + 10 = 3x + 8$
• $5x + 10 ≡ 5(x + 2)$
• $5x + 10 = 5x +2$
• This is an equation because the expression on the left of the equals sign cannot be rearranged to give the equation on the right. The solution to the equation is $$x = -1$$.
• This is an identity because when you expand the bracket on the right of the identity sign, it gives the same expression as on the left of the identity sign.
• This is an equation because the expression on the left of the equals sign cannot be rearranged to give the equation on the right. There is no solution for this equation – no matter what value of $$x$$ is substituted into the equation, the expression on the left will never have the same value as the expression on the right.