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Chapter 2: Problem 12

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

Short Answer

Expert verified
The solution is \(x = 10\).

Step by step solution

01

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 \( 3x = 2(x+5) \).
02

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 \( 3x = 2x + 10 \).
03

Isolate the Variable x

Subtract \(2x\) from both sides of the equation to get \( 3x - 2x = 10 \). Simplify the left side: \( x = 10 \).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
When solving rational equations like \( \frac{x}{x+5} = \frac{2}{3} \), it鈥檚 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鈥檚 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鈥檚 a straightforward approach:
  • Start with the resulting equation from cross-multiplication. For this problem, it鈥檚 \(3x = 2x + 10\).
  • 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.

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Most popular questions from this chapter

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3 x}{x-1} \cdot \frac{x^{2}-1}{x} \div \frac{x+1}{3} $$

Simplify each expression. In each case, list any values of the variables for which the fractions are not defined. \(\left(\frac{3}{a}+\frac{5}{a^{2}}\right) \div\left(\frac{10}{a}+6\right)+\frac{3}{4}\)

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ 5-\frac{1}{2 y} $$

In \(3-14,\) solve and check each inequality. $$ \frac{a}{4}>\frac{a}{2}+6 $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ a-\frac{3}{2 a} $$
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