Solving linear equations - OCREquations and identities
Forming, using and solving equations are skills needed in many different situations. From balancing accounts to making sense of a mobile phone bill, solving equations is a vital skill.
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 鈮, so the example could be written as \(2x + 3x 鈮 5x\).
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 鈥 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.