Problem 18 - Algebra 1

Question

Solve the proportion. Check for extraneous solutions. $$\frac{4}{2 x}=\frac{7}{3}$$

Step-by-Step Solution

Answer

The solution to the proportion is \(x = 0.857\).

Step 1: Cross-Multiply

Cross-multiply the two fractions which means multiply the numerator of the first fraction by the denominator of the second fraction and the denominator of the first fraction by the numerator of the second fraction. The equation becomes \(4 * 3 = 2x * 7\), which simplifies to \(12 = 14x\).

Step 2: Solve the Equation

Divide both sides of the equation by 14 to solve for x. So, \(x = 12/14\) which simplifies to \(x = 0.857\).

Step 3: Check for Extraneous Solutions

Substitute the value of x obtained in the original proportion to check if it is a correct solution or an extraneous solution. Plugging in \(0.857\) for x, we get \(\frac{4}{2 * 0.857} \approx \frac{7}{3}\). Both sides of the equation are approximately equal, verifying our solution.

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool that allows you to solve proportions easily. A proportion is an equation that states two ratios are equal. In simple terms, if you have a fraction on each side of the equation, you can solve the proportion using cross-multiplication.

Here's how it works: multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction. For the equation \(\frac{4}{2x} = \frac{7}{3}\), you would do the following steps:

• Multiply 4 by 3, getting 12.
• Multiply 2x by 7, getting 14x.

Now, you have a simpler equation: \(12 = 14x\). This step transforms the problem into one involving basic algebra, making it much easier to solve the unknown variable \(x\).

Extraneous Solutions

When solving equations, especially involving proportions, you need to be mindful of extraneous solutions. These are values that emerge during the solving process but do not satisfy the original equation.

An extraneous solution often appears when you're dealing with fractions or square roots because of the operations used to eliminate these terms. To detect if a solution is extraneous, plug the result back into the original equation. If both sides of the equation match up or are nearly equal, your solution is likely correct.

In our proportion \(\frac{4}{2x} = \frac{7}{3}\), after finding that \(x = 0.857\), we substitute it back. Both sides approximate the same value, confirming it’s not an extraneous solution. Always make sure to cross-check, as this verification step is crucial in confirming your results.

Equation Solving

Equation solving involves finding the unknown variable that makes the equation true. After cross-multiplying, you'll usually end up with a linear equation, which is much easier to tackle.

From our cross-multiplication, we derived \(12 = 14x\). To solve for \(x\), divide both sides by 14:

• \(x = \frac{12}{14}\)

This simplifies to \(x = 0.857\). Breaking down the problem into smaller steps, as we did here, simplifies the process and ensures accuracy.

Remember to solve equations systematically:\

• Isolate the variable.
• Simplify each step as much as possible.
• Check your solution by substituting back into the original equation.

Following these steps ensures that your solution is both correct and complete.