Problem 32 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

x = -5

Step 1: Set Up the Equation

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

Step 2: Cross-Multiply

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

Step 3: Distribute the 5 on the Right Side

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

Step 4: 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: \[ -2x = 10 \]

Step 5: Solve for x

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

Key Concepts

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.\
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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:\