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Magazine Chapter 7: Problem 7
Solve each proportion. $$ \frac{9}{5}=\frac{12}{3 x+2} $$
Short Answer
Expert verified
The value of \( x \) is \( \frac{14}{9} \).
Step by step solution
01
Understand the Proportion
The proportion given is \( \frac{9}{5} = \frac{12}{3x+2} \). This is a statement where two ratios are equal. Our task is to solve for \( x \).
02
Cross Multiply
In a proportion, you can cross-multiply to solve for the unknown. Cross multiplying \( \frac{9}{5} = \frac{12}{3x+2} \) means multiplying diagonally: \( 9(3x + 2) = 5(12) \).
03
Distribute Terms
Distribute the 9 on the left across the terms inside the parenthesis: \( 9 \times 3x + 9 \times 2 = 5 \times 12 \). Simplifying, we have \( 27x + 18 = 60 \).
04
Isolate the Variable Term
Subtract 18 from both sides to isolate the term with \( x \): \( 27x + 18 - 18 = 60 - 18 \). This simplifies to \( 27x = 42 \).
05
Solve for x
Divide both sides by 27 to solve for \( x \): \( x = \frac{42}{27} \). Simplifying the fraction, \( x = \frac{14}{9} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross Multiplication
Cross multiplication is a technique used to solve proportions, which are equations that assert two ratios are equal. In the given exercise, the proportion is expressed as \( \frac{9}{5} = \frac{12}{3x+2} \). Here, the method of cross multiplication allows us to eliminate the fractions by multiplying the numerator of each ratio by the denominator of the other ratio.
- Start by identifying the numerator and denominator on both sides: 9 and 5 on one side, and 12 and \( 3x + 2 \) on the other.
- For cross multiplication: multiply 9 by \( 3x + 2 \) and multiply 5 by 12.
Isolating Variables
Isolating the variable is a crucial step in solving equations. To isolate a variable means to get the variable alone on one side of the equation, helping us find its value.
- After performing cross multiplication, the equation becomes \( 27x + 18 = 60 \).
- We need to free \( x \) from the other terms interfering with it.
- Begin by clearing constants from the side with the variable. Subtract 18 from both sides, resulting in \( 27x = 42 \).
Distributive Property
The distributive property is vital for expanding expressions and simplifying calculations. When you distribute, you're effectively applying multiplication over addition or subtraction inside parentheses.
- From our original equation, after cross multiplying, we have \( 9(3x + 2) \).
- Apply the distributive property by multiplying 9 by each term inside the parentheses separately: \( 9 \times 3x \) and \( 9 \times 2 \).
- This results in the expanded equation: \( 27x + 18 \).
Fraction Simplification
Simplifying fractions is essential to presenting your final answer cleanly. After isolating the variable, the final equation we derive gives us \( x = \frac{42}{27} \). This fraction isn't in its simplest form yet, so we need to simplify it.
- Find the greatest common divisor (GCD) of the numerator and the denominator. For 42 and 27, the GCD is 3.
- Divide the numerator and the denominator by their GCD: \( \frac{42}{27} \div 3 = \frac{14}{9} \).
