Problem 80 - 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. $$\frac{5}{y}=\frac{10}{?}$$

Step-by-Step Solution

Answer

The expression that makes the rational expressions equivalent is \(2y\).

Step 1 - Understand the equivalence of rational expressions

Two rational expressions are equivalent if their cross products are equal. In other words, for the given equation \(\frac{5}{y} = \frac{10}{?}\), the following must hold true: \(5 \times ? = 10 \times y\).

Step 2 - Set up the cross-multiplication equation

Write the equation from the cross multiplication: \(5 \times ? = 10 \times y\).

Step 3 - Solve for the unknown

To find the value of \( ? \), isolate \( ? \) on one side of the equation. Divide both sides by 5: \[ ? = \frac{10 \times y}{5} \]

Step 4 - Simplify the expression

Simplify \(\frac{10 \times y}{5}\) to \(2y\). Therefore, \(\frac{5}{y} = \frac{10}{2y}\).

Key Concepts

Rational Expressions

Rational expressions are like fractions, but instead of just numbers, they can include variables in the numerator, denominator, or both. They look like this: \(\frac{A}{B}\), where both A and B are polynomials. In our exercise, we see the rational expression \(\frac{5}{y}\). To solve problems involving rational expressions, you often need to find equivalents or simplify them.
One key rule with rational expressions is that the denominator should never be zero because division by zero is undefined. This helps us understand the constraints and ensures that our solutions will be valid. When dealing with these expressions, our goal is usually to make adjustments or find equivalent forms to solve for an unknown value.

Cross Multiplication

Cross multiplication is a method used to solve equations that involve two rational expressions set equal to each other. Here's the idea: if you have \(\frac{a}{b} = \frac{c}{d}\), then the cross products \(a \times d = b \times c\). This technique is especially helpful for solving equations with fractions because it lets us get rid of the fractions and work with whole numbers or expressions instead.
Let's return to our initial problem: \(\frac{5}{y} = \frac{10}{?}\). Applying cross multiplication here, we get: \(5 \times ? = 10 \times y\). This setup turns our original equation with variables in the denominator into a simpler form where we can easily solve for the missing value.

Solving for Unknowns

Solving for unknowns in an equation usually involves isolating the variable you're trying to find. After setting up the cross-multiplication equation from our problem \(5 \times ? = 10 \times y\), we need to isolate the question mark (?), our unknown.
We do this by dividing both sides of the equation by 5: \ ? = \frac{10 \times y}{5} \ This simplifies to: \ ? = 2y \ After simplifying, we find that \(\frac{5}{y} = \frac{10}{2y}\), meaning the value of our original unknown is \(2y\). This process underscores the importance of algebraic manipulation skills in finding solutions to equations involving rational expressions.
Remember: the key is to perform the same operation on both sides of the equation to maintain balance and eventually isolate the variable you're solving for.