Question

Transpose each of the following formulae to make the given variable the subject: (a) $x=\frac{c}{y}$, for $y$ (b) $x=\frac{c}{y}$, for $c$ (c) $k=\frac{2n+5}{n+3}$, for $n$ (d) $T=2\pi \sqrt{\frac{R-L}{g}}$, for $R$

Short answer

Question: Rewrite each of the following equations to make the given variable the subject: a) Make y the subject in the equation $$xy=c$$. b) Make c the subject in the equation $$xy=c$$. c) Make n the subject in the equation $$\frac{k}{n+3}=2n+5$$. d) Make R the subject in the equation $$2\pi \sqrt{\frac{R-L}{g}}=T.$$

Step-by-step solution

(a) Make y the subject

To make y the subject, we need to isolate y. To do this, we start by multiplying both sides of the equation by y. This will give us: $$xy=c$$ Now, we divide both sides by x to isolate y: $$y=\frac{c}{x}$$

(b) Make c the subject

To make c the subject, we need to isolate c. Since c is already in the numerator, we simply multiply both sides of the equation by y to obtain: $$c=xy$$

(c) Make n the subject

To make n the subject, we first need to clear the fractions. To do this, we multiply both sides of the equation by $(n+3)$, which gives: $$k(n+3)=2n+5$$ Next, we distribute k on the left side of the equation: $$kn+3k=2n+5$$ Now, we need to isolate n. To do this, we first subtract 2n from both sides to have all n terms at one side: $$kn-2n+3k=5$$ Now, we factor out n: $$n(k-2)=5-3k$$ Finally, we divide both sides by $(k-2)$ to isolate n: $$n=\frac{5-3k}{k-2}$$

(d) Make R the subject

To make R the subject, we start by isolating the square root on one side. We do this by dividing both sides by $2\pi$: $$\sqrt{\frac{R-L}{g}}=\frac{T}{2\pi }$$ Now, we square both sides of the equation to eliminate the square root: $$\frac{R-L}{g}={\left(\frac{T}{2\pi }\right)}^2$$ Next, we multiply both sides by g to clear the denominator: $$R-L=g{\left(\frac{T}{2\pi }\right)}^2$$ Finally, we add L to both sides to isolate R: $$R=g{\left(\frac{T}{2\pi }\right)}^2+L$$

Key concepts

Transposing equations

Transposing equations is a crucial skill in algebra that involves rearranging the components of an equation in order to isolate a specific variable. It is akin to solving a puzzle, where the goal is to "move" parts of the equation around legally using operations like addition, subtraction, multiplication, and division. This manipulation is carefully done to maintain the equality of both sides. For example, if you are given an equation with fractions, you might need to multiply both sides to clear those fractions first.
Steps to transpose an equation include:

• Identify the variable to isolate.
• Use opposite operations to "move" terms to different sides of the equation.
• Keep the equation balanced by performing the same operations on both sides.
• Simplify the equation as you isolate the desired variable.

Transposing is not just a mechanical process; understanding why each step is taken helps you to handle complex equations more confidently. For instance, making $y$ the subject in $x=\frac{c}{y}$ involves multiplying both sides by $y$ to shift $y$ from the denominator, and then dividing by $x$ to isolate $y$ completely.

Subject of a formula

When we talk about making a variable the "subject" of a formula, we mean rearranging the equation so that this variable stands alone on one side—usually the left side. This is often done in science and engineering to express one quantity in terms of others. It makes formulas more user-friendly and helps in deriving one variable based on known values of other variables.
To make a variable the subject, follow this method:

• Identify the variable you need to make the subject.
• Rearrange the equation such that the variable is isolated on one side.
• Perform inverse operations needed to get rid of other terms around the variable.

For example, making $R$ the subject of the formula $T=2\pi \sqrt{\frac{R-L}{g}}$, involves steps like dividing by $2\pi$, squaring both sides to remove the square root, and manipulating constants using addition or multiplication to isolate $R$. This breakdown helps in expressing one physical quantity solely in terms of others, facilitating easier calculations in practical scenarios.

Isolating variables

Isolating variables is a fundamental process in algebra that makes solving and understanding equations easier. The aim is to identify the variable you are interested in and perform mathematical operations to have it by itself on one side of the equation. This typically involves simplifying the equation and using inverse operations.
To effectively isolate a variable, you should:

• Identify which variable you want to isolate.
• Perform operations such as addition, subtraction, multiplication, or division to eliminate coefficients or other variables.
• Simplify the equation step by step, tracking changes as you proceed.
• Ensure that whatever operation you perform on one side of the equation is done equally to the other side to maintain balance.

For instance, when isolating $n$ in the equation $k=\frac{2n+5}{n+3}$, clear fractions by multiplying through by $n+3$, then move the terms involving $n$ to one side and non-related terms to the other, before factoring and simplifying to have $n$ by itself. Isolating variables often provides deeper insights into how varying one term impacts the whole equation.