Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-20,\) solve each equation and check. $$ \frac{x}{x+5}=\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = 10\).

Step 1: Set the Equation for Cross-Multiplication

We start with the equation \( \frac{x}{x+5} = \frac{2}{3} \). Since we have a proportion, we can use cross-multiplication to solve it. This leads to the equation \( 3x = 2(x+5) \).

Step 2: Distribute the 2 on the Right Side

Distribute the 2 to both terms inside the parentheses on the right side: \( 2(x+5) = 2x + 10 \). Therefore, our equation now is \( 3x = 2x + 10 \).

Step 3: Isolate the Variable x

Subtract \(2x\) from both sides of the equation to get \( 3x - 2x = 10 \). Simplify the left side: \( x = 10 \).

Step 4: Check the Solution

To ensure the solution is correct, substitute \(x = 10\) back into the original equation: \( \frac{10}{10+5} = \frac{2}{3} \). Calculate the left side: \( \frac{10}{15} = \frac{2}{3} \). Simplify \( \frac{10}{15} \) to get \( \frac{2}{3} \), confirming that both sides are equal.

Key Concepts

Cross-Multiplication

When solving rational equations like \( \frac{x}{x+5} = \frac{2}{3} \), it’s essential to understand cross-multiplication. Rational equations often come in the form of proportions, where two ratios are set equal to each other. Cross-multiplication is the technique that allows us to solve these equations easily.

Here’s how it works for our example:

• Begin with the equation \( \frac{x}{x+5} = \frac{2}{3} \).
• Cross-multiply: Multiply the numerator of each fraction by the denominator of the opposite fraction.
• This results in the equation \(3x = 2(x+5)\), transforming the problem into a linear equation.

With cross-multiplication, you effectively eliminate the fractions, making it simpler to isolate variables and solve for \(x\). Always ensure each side of the equation multiplies correctly to avoid mistakes.

Equation Check

Checking your work is a crucial step in solving equations, especially rational ones. It ensures that your solution satisfies the original equation. Here's how to carry out an effective equation check:

• Substitute the found value back into the original equation.
• In our case, substitute \(x = 10\) into \( \frac{x}{x+5} = \frac{2}{3} \).
• This changes the equation to \( \frac{10}{10+5} = \frac{2}{3} \).
• Simplify the left-hand side to get \( \frac{10}{15} = \frac{2}{3} \).

If both sides are equal after simplification, as they are here, it verifies the solution \(x = 10\) is correct. This step helps catch algebra errors and confirms the solution makes sense in the context of the problem.

Isolating Variables

After applying cross-multiplication in rational equations, the next critical step is isolating the variable you are solving for. Here’s a straightforward approach:

• Start with the resulting equation from cross-multiplication. For this problem, it’s \(3x = 2x + 10\).
• Isolate \(x\) by getting all terms containing \(x\) on one side. Subtract \(2x\) from both sides to simplify.
• This simplifies to \(3x - 2x = 10\), which further reduces to \(x = 10\).

Always perform similar operations to maintain balance in the equation, leading to a clean solution for the variable. Proper isolation of variables ensures clarity and precision in solving complex problems.