Problem 77 - Algebra for College Students

Question

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$5=\frac{10}{?}$$

Step-by-Step Solution

Answer

The expression that makes the rational expressions equivalent is 2.

Step 1: Understand the Equality

The given equation is: \[ 5 = \frac{10}{?} \]The goal is to find an expression that makes this equality hold true. Essentially, we need to determine the value of the denominator that makes the fraction on the right side equal to 5.

Step 2: Isolate the Unknown

To isolate the question mark (denoted as \( x \)), rewrite the equation to solve for \( ? \): \[ 5 = \frac{10}{x} \]This implies that finding \( x \) involves basic algebraic manipulation.

Step 3: Solve for the Unknown

To find \( x \), multiply both sides of the equation by \( x \) to get rid of the denominator:\[ 5x = 10 \]Next, solve for \( x \) by dividing both sides of the equation by 5:\[ x = \frac{10}{5} \]

Step 4: Simplify the Expression

Simplify the expression to find the value of \( x \): \[ x = 2 \]So, the expression that should replace the question mark is \( 2 \).

Key Concepts

equivalent expressions

In algebra, we often come across rational expressions and need to determine if they are equivalent. In the given exercise, the goal is to find a value for the question mark that makes the two sides of the equation equal. This involves understanding that both sides of the equation should represent the same value.

For example, if we start with the equation \[ 5 = \frac{10}{?} \], we need to find what the denominator should be to make this true. By figuring this out, we make both expressions equivalent because they would produce the same value when evaluated.

basic algebraic manipulation

Basic algebraic manipulation involves using mathematical operations to simplify or solve equations. In this exercise, we need to isolate the unknown variable (question mark) to find its value. This is done through a series of steps:

• Isolate the unknown: Rewrite the equation to focus on the unknown. For example, \[ 5 = \frac{10}{x} \].
• Eliminate the fraction: Multiply both sides by the denominator to remove the fraction. This gives us \[ 5x = 10 \].
• Solve for the unknown: Divide both sides by 5, resulting in \[ x = 2 \].

These fundamental steps are crucial in solving algebraic equations.

solving for an unknown

Solving for an unknown involves finding the value that satisfies the equation. In simple linear equations like the given example, we perform operations that leave the variable by itself on one side of the equation.

Let's review the steps:

• Start with the equation: \[ 5 = \frac{10}{x} \].
• Multiply both sides by \( x \) to get rid of the denominator: \[ 5x = 10 \].
• Divide both sides by 5 to isolate \( x \): \[ x = 2 \].

By following these steps, we identified \( x = 2 \) as the solution, making \[ 5 = \frac{10}{2} \]. This demonstrates how algebraic techniques can be used to solve for an unknown.