Rearranging Equations – Expanding Brackets Reinforcement Questions

1.

In the equation below, isolate C:

begin mathsize 14px style increment F space equals space left parenthesis R space minus space C right parenthesis space D end style

First, we want to expand the bracket, so we multiply R and C by D:

begin mathsize 14px style increment F space equals space R D space minus space C D end style

Now we want to move RD, but we can’t because it’s being subtracted from. There is a way around this because begin mathsize 14px style R D space minus space C D end style is the same as begin mathsize 14px style negative C D space plus space R D end style, so we can rewrite it as:

begin mathsize 14px style increment F space equals space minus C D space plus space R D end style

Then we move RD across the equals sign by subtracting it from both sides:

begin mathsize 14px style increment F space minus space R D space equals space minus C D space end style

Because begin mathsize 14px style negative C space cross times space D end style is the same as begin mathsize 14px style C space cross times space minus D end style, we can divide both sides by -D to isolate C: 

begin mathsize 14px style C space equals space fraction numerator increment F space minus space R D over denominator negative D end fraction end style

2.

Rearrange the equation below to isolate contains as member:

begin mathsize 14px style fraction numerator negative R over denominator 25 space left parenthesis 2 contains as member space plus space F right parenthesis end fraction space equals space fraction numerator x cubed over denominator increment rho end fraction end style

We start by multiplying out the bracket:

begin mathsize 14px style fraction numerator negative R over denominator 50 space contains as member space plus space 25 F end fraction space equals space fraction numerator x cubed over denominator increment rho end fraction end style

We can’t move -R, so we need to move begin mathsize 14px style 50 contains as member space plus space 25 F end style across by multiplying both sides by it:

begin mathsize 14px style negative R space equals space open parentheses fraction numerator x cubed over denominator increment rho end fraction close parentheses space 50 space contains as member space plus space 25 F end style

Now we can move begin mathsize 14px style open parentheses fraction numerator x cubed over denominator increment rho end fraction close parentheses end style back across by dividing both sides by it:

begin mathsize 14px style fraction numerator negative R over denominator open parentheses begin display style fraction numerator x cubed over denominator increment rho end fraction end style close parentheses end fraction space equals space 50 space contains as member plus space 25 F end style

Now we can subtract 25F from both sides to move that across the equals sign:

begin mathsize 14px style open square brackets fraction numerator begin display style negative R end style over denominator begin display style fraction numerator x cubed over denominator increment rho end fraction end style end fraction close square brackets space minus space 25 F space equals space 50 space contains as member end style

And finally, we can divide both sides by 50 to isolate begin mathsize 14px style contains as member end style:

begin mathsize 14px style contains as member equals space fraction numerator open square brackets begin display style fraction numerator negative R over denominator open parentheses begin display style fraction numerator x cubed over denominator increment rho end fraction end style close parentheses end fraction end style close square brackets space minus space 25 F over denominator space space 50 space end fraction end style

3.

Isolate Y in the following equation:

begin mathsize 14px style left parenthesis Y space plus space x space plus space C right parenthesis squared space equals space fraction numerator z over denominator increment rho end fraction end style

Almost a trick question because you don’t need to expand the bracket; you can just start by taking a square root of both sides of the equation, like so:

begin mathsize 14px style Y space plus space x space plus space C space equals space square root of fraction numerator z over denominator increment rho end fraction end root end style

Next, we can move begin mathsize 14px style x space plus space C end style across the equation by subtracting that from both sides: 

begin mathsize 14px style Y space equals space square root of fraction numerator z over denominator increment rho end fraction end root space minus space left parenthesis x space plus space C right parenthesis end style

4.

Make rho subscript P the subject of the equation below:

begin mathsize 14px style v space equals space 2 over 9 space fraction numerator 3 space left parenthesis space rho subscript P space minus space rho subscript t right parenthesis over denominator mu end fraction space g R squared end style

We can start by moving the begin mathsize 14px style g R squared end style across the equation by dividing both sides by that:

begin mathsize 14px style fraction numerator v over denominator g R squared end fraction space equals space 2 over 9 space fraction numerator 3 space left parenthesis rho subscript P space minus space rho subscript t right parenthesis over denominator mu end fraction space end style

Now we can move the fraction across by multiplying by its reciprocal:

begin mathsize 14px style fraction numerator v over denominator g R squared end fraction space 9 over 2 space equals space fraction numerator 3 space left parenthesis rho subscript P space minus space rho subscript t right parenthesis over denominator mu end fraction space end style

And the begin mathsize 14px style mu end style over by multiplying by that:

begin mathsize 14px style fraction numerator v over denominator g R squared end fraction space 9 over 2 space mu space equals space 3 space left parenthesis rho subscript P space minus space rho subscript t right parenthesis end style

We can’t expand the bracket by multiplying the begin mathsize 14px style rho subscript rho end style and size 14px rho subscript size 14px t by 3 because begin mathsize 14px style rho subscript t end style is being subtracted from begin mathsize 14px style space rho subscript rho end style but we can divide both sides by 3:

begin mathsize 14px style fraction numerator begin display style fraction numerator v over denominator g R squared end fraction end style space begin display style 9 over 2 end style over denominator 3 space end fraction space mu space equals space space rho subscript P space minus space rho subscript t end style

Finally, we can add begin mathsize 14px style rho subscript t end style to both sides to move that over the equals sign isolating begin mathsize 14px style rho subscript rho end style :

begin mathsize 14px style rho subscript P space equals space open parentheses fraction numerator begin display style fraction numerator v over denominator g R squared end fraction 9 over 2 end style over denominator 3 space end fraction space mu close parentheses space plus space rho subscript t end style

5.

Isolate P2 in the following equation:

begin mathsize 14px style Q space equals space fraction numerator K A space left parenthesis P subscript 1 space cross times space P subscript 2 right parenthesis to the power of space 2 end exponent space over denominator mu L end fraction end style

We can start by multiplying both sides by Error converting from MathML to accessible text. to move this across the equation:

begin mathsize 14px style Q mu L space equals space K A space left parenthesis P subscript 1 space cross times space P subscript 2 right parenthesis space squared end style

Now we can move KA by dividing both sides by that:

begin mathsize 14px style fraction numerator Q mu L over denominator K A end fraction space equals space left parenthesis P subscript 1 space cross times space P subscript 2 right parenthesis to the power of space 2 end exponent end style

Now, because we have a multiplication in the brackets, we can square both P1 and P2 to expand:

begin mathsize 14px style fraction numerator Q mu L over denominator K A end fraction space equals space P subscript 1 squared space cross times space P subscript 2 squared end style

Now we move begin mathsize 14px style P subscript 1 squared end style by dividing both sides by it:

begin mathsize 14px style fraction numerator begin display style fraction numerator Q mu L over denominator K A end fraction end style over denominator P subscript 1 squared end fraction space equals space P subscript 2 squared end style

And finally, we can take a square root of both sides to isolate P2:

begin mathsize 14px style space P subscript 2 space equals space square root of fraction numerator begin display style fraction numerator Q mu L over denominator K A end fraction end style over denominator P subscript 1 squared end fraction end root end style

Next, we deal with subtraction in equations.

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