Problem 51 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ \frac{5}{x}-\frac{4}{y}=\frac{5}{z} \text { for } x $$

Step-by-Step Solution

Answer

The solution for \( x \) is \( \frac{5yz}{5y + 4z} \).

Step 1: Identify the Target Variable

We need to solve the formula \( \frac{5}{x} - \frac{4}{y} = \frac{5}{z} \) for the variable \( x \). This means we need to isolate \( x \) on one side of the equation.

Step 2: Move Other Terms to the Other Side

To isolate \( x \), add \( \frac{4}{y} \) to both sides to remove it from the left side: \[ \frac{5}{x} = \frac{5}{z} + \frac{4}{y} \]

Step 3: Express as a Single Fraction

Combine the terms on the right into a single fraction by finding a common denominator: \[ \frac{5}{x} = \frac{5y + 4z}{yz} \]

Step 4: Cross-Multiply to Clear Fractions

Cross-multiply to eliminate fractions: \[ 5yz = x(5y + 4z) \]

Step 5: Solve for x

Divide both sides by \( (5y + 4z) \) to get \( x \) by itself:\[ x = \frac{5yz}{5y + 4z} \]

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra. It involves finding the value of a variable that makes the equation true. In the expression \( \frac{5}{x} - \frac{4}{y} = \frac{5}{z} \), our goal is to solve for \( x \). This means we want to rearrange the equation so that \( x \) stands alone on one side, representing its value clearly.
To effectively solve equations, one should be familiar with performing operations like addition, subtraction, multiplication, and division with the terms involved. Each operation in the equation has an inverse, which is utilized to simplify the equation progressively.
For example, when we start with the equation:\( \frac{5}{x} - \frac{4}{y} = \frac{5}{z} \), notice that we must remove terms from one side to begin isolating \( x \). The basic operation principles guide this process, allowing us to manipulate the terms efficiently and correctly. By adding \( \frac{4}{y} \) to both sides of the equation, we can simplify our target equation step by step. Each transformation must maintain the equation's balance—what you do to one side, you must do to the other.

Isolation of Variable

Isolating a variable means rearranging an equation so that the variable is by itself on one side of the equation. This makes it easy to understand what value the variable holds. The procedure for isolating \( x \) in our given equation, \( \frac{5}{x} = \frac{5}{z} + \frac{4}{y} \), involves strategic movement of terms.
To isolate \( x \), you need to first eliminate other fractions or numbers that are on the same side as \( x \). By transferring \( \frac{4}{y} \) to the other side, we clear unwanted terms away from \( x \). Such operations keep the equation balanced while setting the stage for \( x \) to be isolated.
Once the variable is isolated, the value of \( x \) can be expressed in terms of the remaining variables, which simplifies the understanding of the relationship between them within the equation. This side of algebra revolves around logical restructuring to facilitate easier interpretation of an equation's dynamics.

Cross-Multiplication

Cross-multiplication is a powerful algebraic tool useful in equations involving fractions. It helps in "clearing" the fractions to produce a simpler expression that is easier to solve. In the step \( \frac{5}{x} = \frac{5y + 4z}{yz} \), cross-multiplication is used to get rid of the denominator.
The method involves multiplying across the equal sign, creating a product of terms on one side that equals the product of terms on the other. Specifically, from \( \frac{5}{x} = \frac{5y + 4z}{yz} \), we cross-multiply to obtain:\( 5yz = x(5y + 4z) \).
This technique ensures that the algebraic equation can proceed without fractions, which often complicate the manipulation of terms. Continuing from where cross-multiplication was applied, divide both sides by \( (5y + 4z) \) to isolate \( x \), resulting in \( x = \frac{5yz}{5y + 4z} \). Through such algebraic maneuvers, cross-multiplication streamlines solving equations that might seem complex at first glance.