Formulas where Subject Appears Once

Simple Rearranging

Simple rearranging

What are formulas?

• A formula is a rule, definition or relationship between different quantities, written in shorthand using letters (variables)

  • Formulas include an equals sign

• Some examples you should be familiar with are:

  • The equation of a straight line

    • $y = mx + c$

  • The area of a trapezium

    • $\mathrm{Area} = \frac{\left(a + b\right)h}{2}$

  • Pythagoras' theorem

    • $a^{2 }+ b^{2 }= c^2$

How do I rearrange formulas?  

• The letter (variable) that is on its own on one side is called the subject

  • y  is the subject of y = mx + c

• To make a different letter the subject, we need to rearrange the formula

  • This is also called changing the subject

• The method is as follows:

  • First, remove any fractions

    • Multiply both sides by the lowest common denominator

  • Then use inverse (opposite) operations to get the variable on its own

    • This is similar to solving equations

• For example, make $x$ the subject of $\frac{5x+6}{2}=y$

  • First remove fractions

    • Multiply both sides by 2
      $5x+6=2y$

  • Then get $x$ on its own

    • Subtract 6 from both sides
      $5x=2y-6$

    • Divide both sides by 5
      $x=\frac{2y-6}{5}$

  • There may be more than one correct way to write an answer

    • The following are acceptable alternative forms 
      $x=\frac{2y}{5}-\frac{6}{5}$
      $x=\frac{2\left(y-3\right)}{5}$
      $x=0.4\left(y-3\right)$
      $x=0.4y-1.2$

Should I expand brackets?

• If the variable is inside the brackets, you can either:

  • expand the brackets

  • or divide both by the coefficient if it is not zero

• For example, make $x$ the subject of $3(1+x)=y$

  • You can expand and then rearrange

    • $3+3x=y$

    • $\begin{array}{rcl}3x& =& y-3\end{array}$

    • $\begin{array}{rcl}x& =& \frac{y-3}{3}\end{array}$

  • You can divide and then rearrange

    • $1+x=\frac{y}{3}$

    • $x=\frac{y}{3}-1$

  • Both answers are equivalent

    • $\frac{y-3}{3}$ is the same as $\frac{y}{3}-1$

What if I get fractions in fractions?

• Some rearrangements can lead to fractions in fractions 

  • $x=\frac{{\displaystyle \frac{3}{t} }}{2}$

• Either rewrite with a divide sign, $÷$, then use the method of dividing two fractions

  • $x=\frac{3}{t}÷2$
    $$\begin{gathered} x=\frac{3}{t}÷\frac{2}{1} \\ x=\frac{3}{t}\times \frac{1}{2} \\ x=\frac{3}{2t} \end{gathered}$$

• Or multiply top and bottom by the lowest common denominator of the two fractions and cancel

  • $x=\frac{ {\displaystyle \frac{5}{y}} }{{\displaystyle \frac{t}{8}}}$ becomes $x=\frac{ {\displaystyle \frac{5}{y}\times 8y} }{{\displaystyle \frac{t}{8}\times 8y}}=\frac{40}{ty}$

What if I end up dividing by a negative?

• Remember that $\frac{a}{-b}$ (minus below) is the same as $\frac{-a}{b}$ (minus above) and the same as $-\frac{a}{b}$ (minus outside)

  • Though be careful, as $\frac{-a}{-b}$ is $\frac{a}{b}$

• $-2x=y-3$ becomes $x=\frac{y-3}{-2}$ (minus below)

  • This is the same as $x=\frac{-\left(y-3\right)}{2}$ (minus above) or $- \frac{y-3}{2}$ (minus outside)

    • brackets are required for minus above

    • brackets are assumed for minus outside

  • You can also expand the brackets
    $\frac{-\left(y-3\right)}{2}=\frac{-y+3}{2}=\frac{3-y}{2}$