Equations and identities

An equation states that two expressions are equal in value, eg 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 simplify to 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

Are the following an identity or an equation?

  • 5x + 10 = 3x + 8
  • 5x + 10 = 5(x + 2)
  • 5x + 10 = 5x +2
  • 5x + 10 = 3x + 8 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.
  • 5x + 10 = 5(x + 2) is an identity because when you expand the bracket on the right of the equals sign, it gives the same expression as on the left of the equals sign.
  • 5x + 10 = 5x +2 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.