Problem 46 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. Solve \(\frac{m n p}{2 k}=4 k\) for \(p\)

Step-by-Step Solution

Answer

Question: Given the conditional equation \(\frac{m n p}{2 k}=4 k\), solve for \(p\). Answer: \(p = \frac{8k^2}{mn}\)

Step 1: Understand the given equation

We are given the equation: \(\frac{m n p}{2 k}=4 k\). The goal is to solve for \(p\).

Step 2: Multiply both sides by \(2k\)

To get rid of the denominator, we will multiply both sides of the equation by \(2k\): \((2k) \cdot \frac{m n p}{2k} = 4k \cdot (2k)\) On the left side of the equation, the \(2k\) in the numerator and denominator will cancel each other out, leaving us with: \(m n p = 8k^2\)

Step 3: Divide both sides by \(mn\)

Now, we will isolate \(p\) by dividing both sides of the equation by \(mn\): \(\frac{m n p}{mn} = \frac{8k^2}{mn}\) On the left side of the equation, the \("mn"\) in the numerator and denominator will cancel each other out, leaving us with: \(p = \frac{8k^2}{mn}\)

Step 4: Write the final solution

We have successfully isolated \(p\): \(p = \frac{8k^2}{mn}\) This is the final solution for the given conditional equation.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves the strategic use of mathematical operations to modify and rewrite algebraic expressions and equations in a way that makes them easier to understand or solve. It is a critical skill in solving conditional equations where the goal might be to isolate a variable or simplify the expression.

For example, take the equation \(\frac{mn p}{2k}=4 k\). Here, algebraic manipulation is needed to isolate variable \(p\). We start by strategically multiplying both sides by \(2k\) to eliminate the fraction and continue by applying further operations such as division to isolate \(p\). Through such manipulations, we adhere to the core principle of maintaining equality; whatever operation is done to one side is done to the other. This key concept can be used in countless algebraic scenarios, playing a foundational role in simplifying and solving equations.

Isolating Variables

The process of isolating a variable means rearranging an algebraic equation so that the variable we want to solve for stands alone on one side of the equation, typically the left side. This process involves performing a series of algebraic steps that are the inverse operations of what is currently applied to the variable.

Take the previously manipulated equation \(mn p = 8k^2\). Isolating variable \(p\) requires divvying up \(mn\) from it, and we achieve this by dividing both sides by \(mn\). The inverse operation counteracts the multiplication of \(p\) by \(mn\), thus leaving \(p\) by itself. Isolating variables is a fundamental skill in algebra and is vital for finding solutions to equations as well as understanding the relationships between different algebraic entities.

Equation Solving Steps

Equation solving steps are the sequential actions taken to find the value of the unknown in an equation. These steps are methodical and employ various algebraic techniques to ensure a clear path to the solution. The steps usually start with understanding the given problem and then simplifying the equation through algebraic manipulation. As seen in our example \(\frac{mn p}{2k}=4 k\), we started by eliminating the fraction through multiplication.

Then, after simplifying to \(mn p = 8k^2\), we followed with the isolation of \(p\) by dividing both sides by \(mn\). That leads us to the final step of expressing the solution in its simplest form: \(p = \frac{8k^2}{mn}\). These steps empower students to tackle algebraic equations systematically, promoting a thorough comprehension and ability to solve similar problems independently.