Problem 18 - College Algebra with Modeling and Visualization

Question

Solve the equation for \(x\) and then solve it for \(y .\) $$ \frac{x+y}{x-y}=2 $$

Step-by-Step Solution

Answer

\(x = 3y\) and \(y = \frac{x}{3}\).

Step 1: Rewrite the Equation

First, clear the fraction by cross-multiplying. The given equation is \(\frac{x+y}{x-y} = 2\). Multiply both sides by \(x-y\) to get: \(x+y = 2(x-y)\).

Step 2: Expand and Simplify

Distribute the 2 on the right side of the equation: \(x + y = 2x - 2y\). Move all terms involving \(x\) to one side and \(y\) terms to the other side by subtracting \(x\) and adding \(2y\) to both sides: \(y + 2y = 2x - x\). Simplify to get \(3y = x\).

Step 3: Solve for x

From the equation \(3y = x\), we can directly solve for \(x\) as \(x = 3y\).

Step 4: Solve for y Using x = 3y

Substitute \(x = 3y\) back into the simplified result to solve for \(y\). Since \(x = 3y\), rearranging gives \(y = \frac{x}{3}\).

Key Concepts

Equations

In algebra, equations are math statements that assert the equality of two expressions. They are foundational to solving problems, as they allow us to find unknown values. Solving an equation involves determining the value(s) of the variable(s) that make the equation true. For example, in the original exercise, the equation \( \frac{x+y}{x-y} = 2 \) involves finding both \( x \) and \( y \). To find these values, we perform operations that keep the equation balanced, such as adding or subtracting the same quantity from both sides, or dividing each side by the same non-zero number.

Cross-Multiplying

Cross-multiplying is a technique used to eliminate fractions from equations by multiplying the numerator of one fraction by the denominator of the other and vice versa, then setting the products equal. In this exercise, we cross-multiply to clear the fraction in \( \frac{x+y}{x-y} = 2 \). We multiply both sides by \( x-y \), resulting in the equation \( x + y = 2(x-y) \). This step is crucial because it simplifies the equation to a form without fractions, making it easier to solve for the unknowns. Cross-multiplying is widely used when dealing with equations involving ratios or proportions.

Simplifying Expressions

Simplifying expressions involves rewriting them in the most reduced and concise form, often by combining like terms or applying arithmetic operations. After cross-multiplying, we have \( x + y = 2(x-y) \). We simplify the right-hand side by distributing the 2, resulting in \( x + y = 2x - 2y \). Next, we reorganize the equation to isolate terms with \( x \) on one side and terms with \( y \) on the other. By subtracting \( x \) and adding \( 2y \) to each side, we get \( y + 2y = 2x - x \), which simplifies to \( 3y = x \). Simplifying expressions is a fundamental step in solving algebraic equations, especially when dealing with multiple terms.

Variables

Variables are symbols, often represented by letters, used to stand in for unknown or changing values in algebraic expressions or equations. In this exercise, \( x \) and \( y \) are variables we aim to solve for. Using the simplified equation \( 3y = x \), we directly solve for \( x \) as \( x = 3y \). Then, by substituting \( x = 3y \) back into the context of the problem, we solve for \( y \). Rearranging gives us \( y = \frac{x}{3} \). Understanding how to manipulate variables is essential for solving equations and finding specific solutions. It involves recognizing the role each variable plays within the mathematical context and applying properties of equality and arithmetic effectively.