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Chapter 6: Problem 32

Find the solution set to each equation. $$\frac{x}{5}=\frac{x+2}{3}$$

Short Answer

Expert verified
x = -5

Step by step solution

01

Set Up the Equation

First, write down the given equation: \[ \frac{x}{5} = \frac{x+2}{3} \]
02

Cross-Multiply

To get rid of the fractions, cross-multiply so that each side of the equation becomes a whole number: \[ 3x = 5(x + 2) \]
03

Distribute the 5 on the Right Side

Distribute the 5 to both terms inside the parentheses: \[ 3x = 5x + 10 \]
04

Move All x Terms to One Side

Subtract 5x from both sides to bring all x terms to one side of the equation: \[ 3x - 5x = 10 \] Simplifying this, you get: \[ -2x = 10 \]
05

Solve for x

Divide both sides by -2 to isolate x: \[ x = \frac{10}{-2} \] So, \[ x = -5 \]

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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, one of the most important techniques is cross-multiplication. This method helps eliminate fractions by converting them into a simpler, non-fractional form. In the exercise, we started with the equation \(\frac{x}{5} = \frac{x+2}{3}\). To cross-multiply, imagine swapping the denominators and numerators in a diagonal manner: multiply the numerator of the first fraction by the denominator of the second fraction and vice versa.\
\
So, you end up with: \[ 3x = 5(x + 2) \] This transformation simplifies the solving process by dealing with whole numbers rather than fractions.
Distributive Property
The next step involves using the distributive property to eliminate parentheses. The initial equation after cross-multiplication was \[ 3x = 5(x + 2) \] Applying the distributive property means multiplying 5 by both terms inside the parentheses (x and 2).\
So:\

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Most popular questions from this chapter

Perform the indicated operations. Variables in exponents represent integers. $$\frac{m^{3 k}-1}{m^{3 k}+1} \cdot \frac{m^{2 k+1}-m^{k+1}+m}{m^{3 k}+m^{2 k}+m^{k}}$$

Perform the indicated operations. $$\frac{x^{3}+1}{x^{2}-1} \cdot \frac{3 x-3}{x^{3}-x^{2}+x}$$

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