Formulas where Subject Appears Twice

Subject Appears Twicevideo

Subject appears twice

How do I rearrange formulae where the subject appears twice?  

• If the subject appears twice, you will need to factorise at some point

  • E.g. When making $x$ the subject of $x+xy=3-2y$

  • Factorise out $x$on the left to get $x\left(1+y\right)=3-2y$

    • Notice that the subject now only appears once!

  • Then divide both sides by $\left(1+y\right)$ to get $x=\frac{3-2y}{1+y}$

• If the subject appears twice, and any of these are inside a set of brackets, you will need to expand these brackets first

  • E.g. When making $x$ the subject of $c\left(x+2\right)-x=f$

  • Expand the bracket first to $cx+2c-x=f$

  • Keep the $x$ terms on one side $cx-x=f-2c$

  • $x$ can then be made the subject using factorising as above

• If the subject appears on two sides of a formula, you will need to bring those terms to the same side before you can factorise

  • E.g. When making $x$the subject of $3x=y-px$

  • Add $px$ to both sides first to form $3x+px=y$

  • $x$ can then be made the subject using factorising as above

How do I factorise powers of a subject? 

• If the subject appears twice, and both have the same power, you will need to collect these terms together before applying their inverse

  • E.g. When making $x$the subject of $x^2=-p{x}^2+r$

  • Add $p{x}^2$ to both sides first to form $x^2+p{x}^2=r$

  • $x^2$ can then be factorised out $x^2\left(1+p\right)=r$ to give $x^2=\frac{r}{1+p}$

    • Now take plus-or-minus square roots

    • $x=\pm \sqrt{\frac{r}{1+p}}$

• Be careful when square rooting, or cube rooting etc

  • E.g. To make $x$ the subject of $x^3=\frac{t^3+1}{t^3+8}$

    • The whole right hand side must be cube rooted

    • $x=\sqrt[3]{\frac{t^3+1}{t^3+8}}$

    • This cannot be simplified further

    • The right hand side is not equal to $\frac{t+1}{t+2}$, (this is a common error)