Rearranging Formulae Flashcards

All 5 cards in this deck

What are the steps to change the subject of a formula where the new subject appears once?

e.g. make $x$ the subject of $y=\frac{3x+1}{5}$

1. Multiply both sides by the denominator: $5y=3x+1$
2. Undo the addition: $5y-1=3x$
3. Divide by the coefficient: $x=\frac{5y-1}{3}$

What are the steps to make a variable the subject of a formula written as a single algebraic fraction?

e.g. make $y$ the subject of $w=\frac{y^{2}+5}{y^{2}-2}$

1. Multiply both sides by the denominator: $w(y^{2}-2)=y^{2}+5$
2. Expand and collect the $y^{2}$ terms on one side, then factorise: $y^{2}(w-1)=2w+5$
3. Divide, then take the root: $y=\pm\sqrt{\frac{2w+5}{w-1}}$

What are the steps to make a letter the subject of a reciprocal formula?

e.g. make $u$ the subject of $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$

1. Isolate the reciprocal of the new subject: $\frac{1}{u}=\frac{1}{f}-\frac{1}{v}$
2. Write the other side as a single fraction: $\frac{1}{u}=\frac{v-f}{fv}$
3. Take the reciprocal of both sides: $u=\frac{fv}{v-f}$

What are the steps to derive a printed result in a 'show that' rearrangement question?

e.g. four identical frames each use lengths $3x$, $2x$, $x$ and $2y$, and the total length is $300$. Show that $y=\frac{75-6x}{2}$

1. Form the equation from the information: $4(3x+2x+x+2y)=300$
2. Expand and collect like terms: $24x+8y=300$
3. Rearrange to the printed form: $8y=300-24x$, so $y=\frac{75-6x}{2}$

What are the steps to express one variable in terms of another from two given formulae?

e.g. a cone of radius $r$ and height $x$ has four times the volume of a prism of volume $\frac{1}{2}y^{2}x$; find $r$ in terms of $y$

1. Write each formula: cone $=\frac{1}{3}\pi r^{2}x$, prism $=\frac{1}{2}y^{2}x$
2. Form the equation with the multiplier processed: $\frac{1}{3}\pi r^{2}x=2y^{2}x$
3. Cancel the common variable and rearrange: $r=\sqrt{\frac{6y^{2}}{\pi}}$