Problem 44 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \text { Solve } m^{2} n=2 s \text { for } n $$

Step-by-Step Solution

Answer

Question: Solve the conditional equation \(m^2n=2s\) for the variable \(n\). Answer: \(n=\frac{2s}{m^2}\)

Step 1: Write down the given equation

We start with the given equation: $$ m^2n=2s $$

Step 2: Isolate n

To isolate \(n\) on one side, we need to divide both sides of the equation by \(m^2\). This will give us: $$ \frac{m^2n}{m^2}=\frac{2s}{m^2} $$

Step 3: Simplify the equation

The \(m^2\) terms on the left side of the equation will cancel each other out, leaving \(n\) isolated on the left side. The right side of the equation remains the same. After simplification, we get: $$ n=\frac{2s}{m^2} $$ Now, the conditional equation is solved for \(n\) and expressed in terms of the other given variables \(s\) and \(m\).

Key Concepts

Conditional Equations

Conditional equations are expressions that are true only under certain conditions. An important aspect of these equations is that they may not hold for all values of the variables involved.
For instance, the equation given in the exercise, \(m^2n = 2s\), holds true only when specific values of \(m\), \(n\), and \(s\) are substituted that satisfy the equation.
Conditional equations play a critical role in algebra as they help us explore relationships between variables under particular circumstances. This understanding allows us to model real-world problems where conditions dictate the behavior of variables.

Variable Isolation

The process of variable isolation is all about getting a specific variable on its own on one side of the equation. This is crucial in solving algebraic equations as it helps determine what value or expression the variable represents.
In the exercise, we were tasked with solving for \(n\), which means we needed to isolate \(n\) from the equation \(m^2n = 2s\). By dividing both sides of the equation by \(m^2\), the term \(m^2\) cancels out on the left, leaving \(n\) isolated.
Key steps to isolate a variable:

• Determine which variable needs to be isolated.
• Perform inverse operations on both sides of the equation to "remove" other terms.
• Simplify the equation to keep the isolated variable alone.

Through isolation, we can clearly see how one variable depends on others, simplifying the process of understanding and solving equations.

Equation Solving

Equation solving is the method of finding the value of unknown variables that satisfy an equation. In algebra, it usually involves manipulating the equation using arithmetic operations to simplify and solve it for a specific variable.
Solving the conditional equation \(m^2n = 2s\) required us to perform operations that would isolate \(n\). By dividing both sides by \(m^2\), the equation was simplified to \(n = \frac{2s}{m^2}\).
Key concepts in equation solving include:

• Identifying what you are solving for (the unknown variables).
• Using inverse operations such as addition/subtraction and multiplication/division to simplify.
• Checking the solution by substituting back into the original equation to verify accuracy.

Understanding these steps ensures that you can adequately solve a wide range of algebraic equations, thus mastering this essential algebra skill.