Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the formula for the specified variable. $$M=\frac{Q+1}{Q}\( for \)Q$$

Step-by-Step Solution

Answer

Q = \( \frac{1}{M - 1} \)

Step 1: Introduce the Formula

We are given the formula \( M = \frac{Q+1}{Q} \) and need to solve for \( Q \). This means expressing \( Q \) in terms of \( M \).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by \( Q \) to clear the denominator: \( M \cdot Q = Q + 1 \).

Step 3: Rearrange to Isolate Q

Rearrange the equation to isolate terms involving \( Q \) on one side: \( M \cdot Q - Q = 1 \).

Step 4: Factor Out Q

Observe that \( Q \) is common and can be factored out: \( Q(M - 1) = 1 \).

Step 5: Solve for Q

To solve for \( Q \), divide both sides of the equation by \( (M - 1) \): \( Q = \frac{1}{M - 1} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a technique used to reorganize and simplify equations in order to solve for variables. It helps in transforming complicated equations into more manageable forms, making it possible to isolate variables and solve equations efficiently. In algebraic manipulation, we use arithmetic operations such as addition, subtraction, multiplication, and division, as well as factoring.Let's consider the given formula: \[ M = \frac{Q+1}{Q} \]Our goal is to express \( Q \) in terms of \( M \). To manipulate the formula, start by observing that the fraction on the right side can pose challenges. Hence, we rearrange terms and express them in a simpler form, keeping the equality intact. This might involve multiplying both sides by a common denominator or reframing the terms to facilitate easier factorization or simplification. This skill of rewriting equations is essential in solving complex algebra problems. It lays the foundation for isolating terms and bringing clarity to the problem at hand.

Isolation of Variables

The isolation of variables is an essential step in solving algebraic equations. It involves rearranging an equation such that the variable of interest is on one side of the equation, effectively "isolating" it from other terms.In our exercise, after removing the denominator by multiplying everything by \( Q \), we reached:\[ M \cdot Q = Q + 1 \]The goal was to isolate \( Q \). We begin by collecting all terms with the variable \( Q \) on one side. This step leads directly to factoring and simplifying. Reach the expression:\[ M \cdot Q - Q = 1 \]Notice that \( Q \) now appears in both terms on the left side. The next task was to isolate and extract \( Q \) from these terms, making the problem easier to solve. This step sets the stage for the final solution and highlights the importance of the methodical arrangement of terms.

Fraction Elimination

Fraction elimination is crucial when fractions complicate an equation's readability or solve steps. The aim is to remove fractions so the working equation takes a simpler form without fractions.Let's explore how it applies to \[ M = \frac{Q+1}{Q} \]. Fractions can hinder manipulation, particularly where finding a solution involves cross-multiplication or simplification. Here, we multiply both sides by \( Q \) to clear the fraction:\[ M \cdot Q = Q + 1 \]Eliminating the fraction allowed us to work with a more straightforward equation that circle directly into further steps of simplifying and solving for \( Q \). The removal of the fraction enhances clarity and paves the way for easier algebraic manipulation and variable isolation.This approach is frequently applied in algebra to streamline solutions, maintaining simplicity and coherence throughout the solving process. Understanding fraction elimination aids in managing complex algebraic expressions effectively. Keeping equations clear of fractions whenever feasible can significantly ease analysis and problem-solving.