Problem 47 - Intermediate Algebra

Question

Solve each formula for the specified variable. \(P=\frac{Q_{1}}{Q_{2}-Q_{1}}\) for \(Q_{1}\) (from refrigeration/heating)

Step-by-Step Solution

Answer

\(Q_1 = \frac{PQ_2}{1 + P}\)

Step 1: Isolate the Denominator

The given formula is \(P = \frac{Q_1}{Q_2 - Q_1}\). Start by removing the fraction by multiplying both sides of the equation by \(Q_2 - Q_1\), resulting in \( P(Q_2 - Q_1) = Q_1 \).

Step 2: Distribute the Variable

Distribute \(P\) across the terms inside the parentheses: \( PQ_2 - PQ_1 = Q_1 \).

Step 3: Rearrange Terms

Add \(PQ_1\) to both sides to move it from the left to the right: \( PQ_2 = Q_1 + PQ_1 \).

Step 4: Factor Out the Desired Variable

Factor \(Q_1\) from the right side to make it easier to isolate: \(PQ_2 = Q_1(1 + P)\).

Step 5: Isolate the Desired Variable

Divide both sides of the equation by \((1 + P)\) to solve for \(Q_1\): \(Q_1 = \frac{PQ_2}{1 + P}\).

Key Concepts

Refrigeration and Heating

Refrigeration and Heating systems are crucial in maintaining comfortable and safe environments in various settings, from residential to industrial. These systems work by manipulating thermal energy, often involving a transfer of heat or cold to regulate temperatures effectively. In essence, the main function is either to remove heat, as in refrigeration, or provide it, as in heating.
Understanding the mathematical relationships in these processes, such as the equation given in our problem, helps design and optimize such systems. They ensure energy efficiency and cost-effectiveness while maintaining the desired climate conditions. When you encounter variables like \(Q_1\) and \(Q_2\), they typically denote quantities of heat, and how they relate can provide insights into how effectively a system is likely operating.
This translates to real-world applications where precise control over these heat quantities is necessary to maintain system balance and efficiency. It's essential to be able to manipulate equations to isolate variables, as this helps predict outcomes and adjusts inputs accordingly, a crucial aspect of engineering and thermodynamics in refrigeration and heating scenarios.

Solving Equations for a Variable

Solving equations for a specific variable is a foundational skill in algebra. This process involves rearranging the equation so that the variable of interest is isolated on one side of the equation. Let's walk through this with our initial problem as a context.

• Start by addressing any fractions. Multiplying through by the denominator can clear the fraction, simplifying the equation.
• Distribute any terms if necessary to eliminate parentheses and combine like terms.
• Rearrange the equation by moving terms involving the variable to one side. Use addition, subtraction, multiplication, or division as needed to isolate the variable.

In our example, moving terms and factoring enabled us to isolate \(Q_1\), resulting in \(Q_1 = \frac{PQ_2}{1 + P}\). This step-by-step method ensures clarity and prevents errors, a useful approach not only in algebra, but in various scientific calculations and engineering applications.

Factoring in Algebra

Factoring plays a pivotal role in algebra, especially when it comes to simplifying expressions and solving equations. Factoring involves expressing an equation as a product of its factors, which can make manipulation much simpler.
In our problem, the step of factoring involves recognizing common terms and "pulling" them out to simplify the expression. For instance, if you have \(PQ_2 = Q_1 + PQ_1\), notice that \(Q_1\) appears in both terms on one side of the equation, i.e., \( Q_1(1 + P) \). By factoring out \(Q_1\), the equation becomes easier to solve.
Understanding factoring not only helps in solving equations but also in decomposing complex problems into simpler parts, facilitating easier computation and often better understanding of the underlying mathematical relationships.