Question

Make \(n\) the subject of the formula $$ J=\frac{n E}{n L+m} $$

Short answer

Question: Given the formula $$\frac{nE}{nL + m} = J$$, find n in terms of E, L, J, and m. Answer: After making n the subject of the formula, we get: $$n = \frac{Jm}{E - JL}$$

Step-by-step solution

Clear the denominator

To clear the denominator, multiply both sides of the equation by \((nL + m)\): $$ J(nL + m) = \frac{nE}{nL + m}(nL + m) $$ This simplifies to: $$ J(nL + m) = nE $$

Distribute J

Distribute J to both terms within the parentheses: $$ JnL + Jm = nE $$

Isolate n terms

Move the \(JnL\) term to the right side of the equation to have all n terms on the right side: $$ Jm = nE - JnL $$

Factor out n

Factor n out on the right side of the equation: $$ Jm = n(E - JL) $$

Solve for n

Divide both sides of the equation by \((E - JL)\) to solve for n: $$ n = \frac{Jm}{E - JL} $$ Now, \(n\) is the subject of the formula.

Key concepts

Isolating Variables

Isolating variables is a crucial skill in algebra that forms the basis for solving equations. The idea is to rearrange the equation until the variable of interest is by itself on one side. This process involves a series of algebraic manipulations like adding, subtracting, multiplying, or dividing both sides of the equation to maintain balance.
To start isolating a variable, first determine the current position and operations surrounding the variable. In the problem, to isolate \(n\), we needed to get it on one side of the equation by itself. We began by eliminating the denominator, \(nL + m\), by multiplying both sides which simplified our process.
Key steps include:

• Clearing unwanted terms from around the variable
• Using inverse operations to move terms
• Factoring or simplifying expressions containing the variable

Remember, maintaining equality during each step is essential, ensuring that what is done to one side of the equation is equally done to the other.

Equations

An equation is a mathematical statement that asserts the equality of two expressions. It is composed of two expressions, often an expression on the left set equal to one on the right. Understanding how equations work is essential for solving problems and determining relationships between variables.
In our example, we began with the equation \(J=\frac{n E}{n L+m}\). The goal was to solve for \(n\), indicating it as the subject of the formula. This involved manipulating the original equation in various steps until \(n\) was isolated.
The process:

• Recognized the equation involves fractions and began by clearing the fraction.
• Used operations to simplify each part, maintaining equation balance.
• Manipulated the equation until the desired variable was clearly isolated.

Understanding equations helps us see how changing one quantity affects another, emphasizing the importance of each step in maintaining the integrity of the equation.

Mathematical Proof

Mathematical proof is a series of logical steps used to demonstrate the truth of a mathematical statement. While solving for a variable within an equation doesn't constitute a full proof in a formal sense, the approach and logic involved share similarities with proving math concepts.
Each manipulation in our process served as a "mini-proof", ensuring that each step logically followed from the previous. The steps:

• Verify the correctness at each stage of algebraic manipulation.
• Ensure operations like multiplying and dividing don't introduce errors, such as division by zero.
• Check the final arrangement matches the problem's requirements.

Using clear and logical steps helps verify the solution's correctness, akin to constructing a mathematical proof in more formal mathematical studies.