Problem 10 - Algebra 1

Question

Solve the proportion. Check for extraneous solutions. $$\frac{6}{x}=\frac{5}{3}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3.6\).

Step 1: Cross-multiply

Multiply the numerator of the first fraction by the denominator of the second one, and the denominator of the first by the numerator of the second. So,\(6*3 = 5*x\) will yield \(18 = 5x\).

Step 2: Solving for the unknown

To solve for \(x\), divide each side of the equation by 5. Therefore, \(x = 18/5 = 3.6\).

Step 3: Check for Extraneous Solution

Substitute the value of \(x\) back into the original equation to check for accuracy. Hence, \(\frac{6}{3.6} = \frac{5}{3}\), simplifying both sides we obtain \(1.67 \approx 1.67\). The left-hand side equals the right-hand side, so the solution is correct.

Key Concepts

Cross-Multiplication Method

Understanding the cross-multiplication method is the key to solving proportions like \( \frac{6}{x} = \frac{5}{3} \). Cross-multiplying is a technique where you multiply the numerator (top part) of one fraction by the denominator (bottom part) of another. Then, you do the same with the remaining numerator and denominator. This sets up an equation without fractions, which is often easier to solve.

In this exercise, you cross-multiply to create the equation:

• Numerator of the first fraction \(6\) multiplied by the denominator of the second fraction \(3\): \(6 \times 3 = 18\).
• Denominator of the first fraction \(x\) multiplied by the numerator of the second fraction \(5\): \(5 \times x = 5x\).

Now the equation is \(18 = 5x\), a simple algebraic equation that you can solve for \(x\) by dividing both sides by 5. This step is straightforward once the proportion is set up using cross-multiplication.

Extraneous Solutions

Extraneous solutions are results that emerge from the solving process but don't satisfy the original equation. It's crucial always to replace the solution back into the original equation to ensure it holds true.

When solving the proportion \( \frac{6}{x} = \frac{5}{3} \), we found that \( x = 3.6 \). To check for an extraneous solution:

• Substitute \( x = 3.6 \) back into \( \frac{6}{x} \), obtaining \( \frac{6}{3.6} \).
• Simplify \( \frac{6}{3.6} = 1.67 \) and check if it equals \( \frac{5}{3} = 1.67 \).

Both values match closely, confirming the solution is accurate and not extraneous. This reaffirms the need for this checking step, as skipping it might lead to accepting an incorrect or non-valid solution.

Algebraic Equations

Algebraic equations are mathematical statements of equality involving variables. They consist of two expressions connected by an equal sign. Solving equations involves finding the value of the variable that makes the equation true. In this context, the equation formed by cross-multiplying the given proportion \( \frac{6}{x} = \frac{5}{3} \) results in the expression \( 18 = 5x \).

Solving this equation requires isolating the variable \( x \):

• Divide both sides by the coefficient of \( x \), which is 5, to get \( x = \frac{18}{5} \) or \( x = 3.6 \).

This step-by-step process simplifies the problem and makes it manageable. Understanding how algebraic equations form and how to solve them is crucial to tackling a wide range of mathematical problems similar to this one.