Problem 11 - Algebra 2 and Trigonometry

Question

Solve each proportion for the variable. \(\frac{x}{8}=\frac{6}{24}\)

Step-by-Step Solution

Answer

The value of \( x \) is 2.

Step 1: Cross Multiply

To solve the proportion \( \frac{x}{8} = \frac{6}{24} \), use the cross multiplication method. Multiply the numerator of one fraction by the denominator of the other: \( x \times 24 \) and \( 6 \times 8 \). This gives the equation \( 24x = 48 \).

Step 2: Solve for x

Next, isolate \( x \) by dividing both sides of the equation by 24: \( \frac{24x}{24} = \frac{48}{24} \). This simplifies to \( x = 2 \).

Step 3: Verify the Solution

To ensure the solution is correct, substitute \( x = 2 \) back into the original proportion: \( \frac{2}{8} = \frac{6}{24} \). Simplifying both sides, we have \( \frac{1}{4} = \frac{1}{4} \), which confirms the solution is correct.

Key Concepts

Cross Multiplication

Cross multiplication is a technique used to solve proportions like \( \frac{x}{8} = \frac{6}{24} \). It involves multiplying diagonally across the equal sign to simplify the problem. Here’s how it works: you multiply the numerator (top number) of one fraction by the denominator (bottom number) of the other fraction. This creates an equation without fractions, often making it easier to solve. In this case:

• Multiply \( x \) (numerator on the left) by 24 (denominator on the right): \( x \times 24 \).
• Multiply 6 (numerator on the right) by 8 (denominator on the left): \( 6 \times 8 \).

This results in the equation \( 24x = 48 \), a key step in solving the proportion. By removing fractions, cross multiplication helps in focusing on the core numerical values, streamlining the solving process.

Equation Solving

Equation solving is the process of finding the unknown variable that satisfies the equation. After establishing the cross-multiplying equation \( 24x = 48 \), the next step is to isolate the variable, \( x \). This typically involves performing inverse operations to both sides of the equation to keep it balanced. Here’s how it proceeds:

• Divide every term by 24 to isolate \( x \): \( \frac{24x}{24} = \frac{48}{24} \).
• Simplify the right side: \( x = 2 \).

Through division, we effectively turn the complex part of the equation into a simpler statement. It showcases the power of inverse operations, where dividing by a number counteracts multiplication. This balance leads you directly to the solution, \( x = 2 \), showing the elegance and practicality of mathematical reasoning.

Verification of Solution

Verification of solution is a critical step to ensure the accuracy of your work. Once we found the answer \( x = 2 \), we must check if it holds true in the original equation \( \frac{x}{8} = \frac{6}{24} \). Here's how you can verify:

• Substitute \( x = 2 \) back into the original proportion: \( \frac{2}{8} \).
• Simplify both sides of the equation to confirm equality.
• The left-hand side simplifies to \( \frac{1}{4} \).
• The right-hand side, \( \frac{6}{24} \), also simplifies to \( \frac{1}{4} \).

Both sides equal \( \frac{1}{4} \), proving that \( x = 2 \) is indeed the correct solution. This step not only reassures the correctness of our solution but also builds confidence in our ability to carry out precise calculations. Verifying solutions is a must, especially in complex problems.