Question

If \(x / 2+y / 3=x / 3+y / 2\), then \(x+y\) is: A. \(2 x 2 y\) B. 0 C. \(6 x\) D. \(2 x\) E. Cannot be determined

Short answer

The correct answer is D. \( 2x \).

Step-by-step solution

Rewrite the Equation

Given the equation \( \frac{x}{2} + \frac{y}{3} = \frac{x}{3} + \frac{y}{2} \). Rewrite the equation so that all terms are on one side of the equation. This gives: \( \frac{x}{2} + \frac{y}{3} - \frac{x}{3} - \frac{y}{2} = 0 \).

Find a Common Denominator

To simplify the equation, find a common denominator for the fractions. The common denominator for 2 and 3 is 6. Thus: \( \frac{3x}{6} + \frac{2y}{6} - \frac{2x}{6} - \frac{3y}{6} = 0 \).

Combine Like Terms

Combine the fractions by adding and subtracting the numerators over the common denominator: \( \frac{3x + 2y - 2x - 3y}{6} = 0 \).

Simplify the Numerator

Simplify the numerator: \( 3x - 2x + 2y - 3y = x - y \). This results in: \( \frac{x - y}{6} = 0 \).

Solve for x - y

Since the fraction equals zero, the numerator must be zero: \( x - y = 0 \). Solving for x and y, we get: \( x = y \).

Find x + y

Since \( x = y \), the expression for \( x + y \) becomes: \( x + x = 2x \). Therefore: \( x + y = 2x \).

Pick the Correct Answer

Comparing with the given options, the correct answer is D. \( 2x \).

Key concepts

Equation Simplification

Simplifying equations is crucial in algebra. It helps in solving problems by making them more manageable. In the given exercise, we started with the equation \(\frac{x}{2} + \frac{y}{3} = \frac{x}{3} + \frac{y}{2}\). To simplify it, we first moved all terms to one side. This resulted in the new equation: \(\frac{x}{2} + \frac{y}{3} - \frac{x}{3} - \frac{y}{2} = 0\). By shifting all terms to one side, we transformed the original problem into a suitable form for finding a common denominator.

Finding a Common Denominator

A common denominator is necessary to add and subtract fractions. For the equation \(\frac{x}{2} + \frac{y}{3} - \frac{x}{3} - \frac{y}{2} = 0\), we needed a common denominator for 2 and 3, the denominators of our fractions. The least common multiple (LCM) of 2 and 3 is 6. So, we rewrote the fractions: \(\frac{3x}{6} + \frac{2y}{6} - \frac{2x}{6} - \frac{3y}{6} = 0\). This conversion made it easier to combine and simplify the terms in the next steps.

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic equations. In our example, we combined the fractions over the common denominator: \(\frac{3x + 2y - 2x - 3y}{6} = 0\). Here, the like terms were 3x and -2x for x terms, and 2y and -3y for y terms. Adding and subtracting these like terms resulted in \(\frac{x - y}{6} = 0\). This simplification process reduces the equation to a more straightforward form.

Solving for Variables

Once we have a simplified equation, the next step is to solve for the variables. From \(\frac{x - y}{6} = 0\), we know that for the fraction to equal zero, the numerator must also be zero. Hence, we set \(x - y = 0\) and solved it to find \(x = y\). With this information, we can easily determine that \(x + y = x + x = 2x\). Thus, the answer is 2x, satisfying the problem's requirements.