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Rearranging Formulas for A Level Maths

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Rearranging Formulas – Introduction

 In A Level Maths, rearranging formulas is crucial when solving equations or manipulating mathematical expressions. By rearranging formulas, you can isolate variables and find their values or express them in terms of other variables. This skill is fundamental in various mathematical concepts and problem-solving scenarios.

  1. Sub Heading: Basic Concepts
     Rearranging formulas involves manipulating equations to isolate a specific variable. The basic concepts include:

Swapping sides: Move terms from one side of the equation to the other by performing the opposite operation. For example, to move a term from the left-hand side to the right-hand side, you would change its sign.

Collecting like terms: Combine similar terms on the same side of the equation to simplify the expression. This step helps in isolating the desired variable.

Applying inverse operations: To isolate a variable, use the inverse operation of the operation applied to it in the original equation. For instance, if the variable is multiplied by a constant, divide both sides of the equation by that constant.

  1. Sub Heading: Examples in Practice
     Let’s consider an example to illustrate rearranging formulas:

Given the formula for the area of a rectangle, A = l × w, where A represents the area, l represents the length, and w represents the width. If we want to find the length, we can rearrange the formula as follows:

Start with the formula: A = l × w
 Divide both sides of the equation by w: A/w = (l × w)/w
 Simplify the right-hand side: A/w = l
 Swap sides to isolate l: l = A/w

Now we have successfully rearranged the formula to express the length (l) in terms of the area (A) and width (w).

Rearranging Formulas - Examples

\begin{aligned} P=\frac{C}{C+4} & \Rightarrow P(C+4)=C \\ & \Rightarrow P C+4 P=C \\ & \Rightarrow P C-C=-4 P \\ & \Rightarrow C(P-1)=-4 P \\ & \Rightarrow C=\frac{-4 P}{P-1} \text { or } \frac{4 P}{1-P} \end{aligned}

Quite often in rearranging formulas questions you can see that the variable that you want to make the subject appear more than once. You need to group these terms together and factorise as shown in the above example. 

\begin{aligned} V=\frac{1}{3} \pi r^2 h & \Rightarrow 3 V=\pi r^2 h \\ & \Rightarrow \frac{3 V}{\pi}=r^2 h \\ & \Rightarrow \frac{3 V}{\pi h}=r^2 \\ & \Rightarrow r= \pm \sqrt{\frac{3 V}{\pi h}} \end{aligned}

\begin{aligned} 2 a+5 c=a f+7 c & \Rightarrow 2 a-a f=2 c \\ & \Rightarrow a(2-f)=2 c \\ & \Rightarrow a=\frac{2 c}{2-f} \text { or } \frac{-2 c}{f-2} \end{aligned}

Rearranging Formulas - More Examples

\begin{array}{r} s=\frac{1}{2} a t^2 \Rightarrow 2 s=a t^2 \Rightarrow t^2=\frac{2 s}{a} \\ \Rightarrow t= \pm \sqrt{\frac{2 s}{a}} \end{array}

\begin{aligned} E=\frac{1}{2} m v^2 \Rightarrow 2 E=m v^2 & \Rightarrow \frac{2 E}{m}=v^2 \\ & \Rightarrow v= \pm \sqrt{\frac{2 E}{m}} \end{aligned}

\begin{aligned} y=\frac{x+3}{x-2} & \Rightarrow y(x-2)=x+3 \\ & \Rightarrow y x-2 y=x+3 \\ & \Rightarrow y x-x=3+2 y \\ & \Rightarrow x(y-1)=3+2 y \\ & \Rightarrow x=\frac{3+2 y}{y-1} \text { or } \frac{-3-2 y}{1-y} \end{aligned}

\begin{aligned} y+5=x(y+2) & \Rightarrow y+5=x y+2 x \\ & \Rightarrow y-x y=2 x-5 \\ & \Rightarrow y(1-x)=2 x-5 \\ & \Rightarrow y=\frac{2 x-5}{1-x} \text { or } \frac{5-2 x}{x-1} \end{aligned}

\begin{aligned} s=u t+\frac{1}{2} a t^2 & \Rightarrow s-u t=\frac{1}{2} a t^2 \\ & \Rightarrow 2\left(s-u()=a t^2\right. \\ & \Rightarrow a=\frac{2(s-u t)}{t^2} \end{aligned}
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Rearranging Formulas – Further Examples  

 

 

\begin{aligned} V=\frac{1}{3} \pi r^2 \sqrt{l^2-r^2} & \Rightarrow 3 V=\pi r^2 \sqrt{l^2-r^2} \Rightarrow \frac{3 V}{\pi r^2}=\sqrt{l^2-r^2} \\ & \Rightarrow\left(\frac{3 V}{\pi r^2}\right)^2=l^2-r^2 \Rightarrow l^2=\left(\frac{3 V}{\pi r^2}\right)^2+r^2 \\ & \Rightarrow l= \pm \sqrt{\left(\frac{3 V}{\pi r^2}\right)^2+r^2} \end{aligned}

\begin{aligned} y=\frac{1-2 x}{x+3} & \Rightarrow y(x+3)=1-2 x \Rightarrow y x+3 y=1-2 x \\ & \Rightarrow y x+2 x=1-3 y \Rightarrow x(y+2)=1-3 y \\ & \Rightarrow x=\frac{1-3 y}{y+2} \end{aligned}

C=\pi d \Rightarrow C_d=\pi d^2

 

\begin{aligned} A=\pi r^2=\pi\left(\frac{d}{2}\right)^2=\frac{\pi d^2}{4} & \Rightarrow 4 A=\pi d^2 \\ & \Rightarrow d^2=\frac{4 A}{\pi} \end{aligned}

 

\begin{aligned} C d=\pi d^2 & \Rightarrow C d=\pi\left(\frac{4 A}{\pi}\right) \\ & \Rightarrow C d=4 A \text { so } k=4 \end{aligned}

\begin{aligned} 5 c+9 t=a(2 c+t) & \Rightarrow 5 c+9 t=2 a c+a t \\ & \Rightarrow 5 c-2 a c=a t-9 t \\ & \Rightarrow c(5-2 a)=a t-9 t \\ & \Rightarrow c=\frac{a t-9 t}{5-2 a} \end{aligned}

\begin{aligned} 3(a+4)=a c+5 f & \Rightarrow 3 a+12=a c+5 f \\ & \Rightarrow 3 a-a c=5 f-12 \\ & \Rightarrow a(3-c)=5 f-12 \\ & \Rightarrow a=\frac{5 f-12}{3-c} \text { or } \frac{12-5 f}{c-3} \end{aligned}

\begin{aligned} a=\frac{3 c+2 a}{2 c-5} & \Rightarrow a(2 c-5)=3 c+2 a \\ & \Rightarrow 2 a c-5 a=3 c+2 a \\ & \Rightarrow 2 a c-7 a=3 c \\ & \Rightarrow a(2 c-7)=3 c \\ & \Rightarrow a=\frac{3 c}{2 c-7} \end{aligned}

\begin{aligned} \frac{x}{3 y}=\frac{2 x+1}{y+2} & \Rightarrow x(y+2)=3 y(2 x+1) \\ & \Rightarrow x y+2 x=6 x y+3 y \\ & \Rightarrow 2 x-5 x y=3 y \\ & \Rightarrow x(2-5 y)=3 y \\ & \Rightarrow x=\frac{3 y}{2-5 y} \end{aligned}

Conclusion
 Rearranging formulas is an essential skill in A Level Maths, allowing you to manipulate equations to solve for specific variables or express them in terms of others. By understanding the basic concepts, such as swapping sides, collecting like terms, and applying inverse operations, you can confidently rearrange formulas in various mathematical scenarios. This skill is valuable for tackling complex problems, analysing relationships between variables, and making further calculations based on given formulas.

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