Problem 31 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{10}{x}=\frac{20}{x+20}$$

Step-by-Step Solution

Answer

The solution set is {20}.

Step 1: Set the problem

The given equation is \( \frac{10}{x} = \frac{20}{x+20} \)

Step 2: Cross-multiply

To solve the equation, cross-multiply to get rid of the fractions: \(10(x + 20) = 20x \)

Step 3: Distribute and simplify

Distribute the 10 on the left side: \[ 10x + 200 = 20x \]

Step 4: Isolate the variable

Subtract 10x from both sides to isolate the variable: \[ 10x + 200 - 10x = 20x - 10x \] \[ 200 = 10x \]

Step 5: Solve for x

Divide both sides by 10 to find the value of x: \[ x = \frac{200}{10} \] \[ x = 20 \]

Step 6: Verify the solution

Substitute \( x = 20 \) back into the original equation to check: \[ \frac{10}{20} = \frac{20}{20 + 20} \] \[ \frac{10}{20} = \frac{20}{40} \] \[ \frac{1}{2} = \frac{1}{2} \] Since both sides are equal, the solution is verified.

Key Concepts

cross-multiplication

Cross-multiplication is a technique used to solve rational equations. It involves multiplying the numerator of one fraction by the denominator of the other fraction. This helps to eliminate the fractions and makes the equation easier to solve. In the equation \( \frac{10}{x} = \frac{20}{x+20} \), cross-multiplying gives us: \( 10(x + 20) = 20x \). This step simplifies the process and helps in getting rid of the fractions on both sides. By using cross-multiplication:

• We convert a rational equation into a linear equation.
• The fractions disappear, leaving an equation with no denominators.

This makes the next steps of solving the equation more straightforward.

isolating the variable

Isolating the variable means getting the variable (in this case, \( x \)) by itself on one side of the equation. It's an important step when solving any equation. After cross-multiplying in our example, we get: \[ 10(x + 20) = 20x \]. Distributing on the left side, we get: \[ 10x + 200 = 20x \]. To isolate \( x \), we need to move all terms containing \( x \) to one side and constant terms to the other:

• Subtract \( 10x \) from both sides: \[ 10x + 200 - 10x = 20x - 10x \]
• This simplifies to: \[ 200 = 10x \]
• Finally, divide both sides by 10: \( x = \frac{200}{10} \), which gives \( x = 20 \)

This isolates \( x \) and provides the value that satisfies the equation.

fraction operations

When solving rational equations, understanding how to work with fractions is essential. Here are key points to remember:

In fractions, the numerator (top number) and the denominator (bottom number) must be handled carefully. For instance, in \( \frac{10}{x} \) and \( \frac{20}{x+20} \), the numerators are 10 and 20, while the denominators are \( x \) and \( x+20 \).

Operations with fractions include:

• Addition/Subtraction: Find a common denominator before adding or subtracting the numerators.
• Multiplication: Multiply the numerators together and the denominators together.
• Division: Multiply by the reciprocal of the fraction you're dividing by.

In the case of solving \( \frac{10}{x} = \frac{20}{x+20} \), cross-multiplication simplifies the process by eliminating the fractions, allowing us to focus on solving for \( x \).