Finding the value of an unknown in a formula

To find an unknown in a formula you can either substitute the known values first and then solve the equation, or change the subject first and then substitute. Substituting first is quicker for one value; changing the subject first is better when many values are needed.

Why substituting first is quicker: once the numbers are in, the formula becomes an equation with only one letter, so there is nothing to collect or factorise. A value that makes a denominator zero, or the number under a square root negative, gives no value at all.

Method

1. Substitute the known values, using brackets around negative numbers.
2. Solve the equation for the unknown: undo roots by squaring, clear fractions by multiplying.
3. Check that the answer makes sense in context and gives a defined value.

Example

The mean \(M\) of three marks \(a\), \(b\) and \(c\) is \(M=\dfrac{a+b+c}{3}\). Find \(c\) when \(M=72\), \(a=65\) and \(b=80\).

\(\begin{aligned}72&=\dfrac{65+80+c}{3} \\ 216&=145+c \\ c&=71\end{aligned}\)

A rectangle has length \(l\) cm, breadth \(5\) cm and diagonal \(d=\sqrt{l^2+25}\) cm. Find \(l\) when \(d=13\).

\(\begin{aligned}13&=\sqrt{l^2+25} \\ 169&=l^2+25 \\ l^2&=144\end{aligned}\)

A length is positive, so \(l=12\).

Check: \(-3^2=-9\) but \((-3)^2=9\), so write negative values in brackets when you substitute. Substitute your answer back: \(\sqrt{12^2+25}=13\).