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Chapter 2: Problem 30

Solve each equation. $$ \frac{x}{6}=\frac{18}{4} $$

Short Answer

Expert verified
x = 27

Step by step solution

01

Set Up the Equation

Start with the given equation: \[ \frac{x}{6} = \frac{18}{4} \]
02

Simplify the Right Side

Simplify \( \frac{18}{4} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. \[ \frac{18}{4} = \frac{18 \text{/} 2}{4 \text{/} 2} = \frac{9}{2} \]
03

Cross-Multiply

Cross-multiply to solve for \( x \). \[ x \times 2 = 6 \times 9 \]
04

Calculate the Expression

Multiply both sides of the equation to isolate \( x \). \[ 2x = 54 \]
05

Solve for \( x \)

Divide both sides by 2 to find the value of \( x \). \[ x = \frac{54}{2} = 27 \]

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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 an essential technique for solving rational equations, especially when dealing with fractions on both sides.
It involves multiplying the numerator of one fraction by the denominator of the other and setting them equal.
For example, from the equation \(\frac{x}{6} = \frac{18}{4}\), you multiply x by 4 and 18 by 6.
This translates to: \[ x \times 4 = 18 \times 6 \]
  • Cross-multiplication eliminates the fractions and converts the equation into a simpler form.
  • It helps in isolating the variable without extensive manipulation.
Remember, you can only cross-multiply when both sides of the equation have one fraction each.
Simplifying Fractions
Isolating the variables means getting the unknown alone on one side of the equation.
This makes solving equations straightforward.
After simplifying and cross-multiplying from our example, we get: \(\frac{2x}{2} = \frac{54}{2} \)solving isolates x.
  • This step straightforward calculation towards finding the solution.
  • Division cancels out coefficients, providing the value.
  • Isolate by simplifying fractions and coefficients.
  • Utilize variable when necessary.

This will allow you to efficiently solve most rational equations.
Greatest Common Divisor
Finding the Greatest Common Divisor (GCD) is crucial in simplifying fractions.
The GCD is the largest number dividing both the numerator and denominator without leaving a remainder.
In our example: \[ \frac{18}{4} \]
The factors of 18 are 1, 2, 3, 6, 9, 18 and factors of 4 are 1, 2, 4.
The highest common factor is 2, making it the GCD.
  • Using GCD simplifies fractions to their smallest terms.
  • Helps in further straightforward calculations.
You can always find it using prime factorization or Euclidean algorithm for more complex fractions.
This makes handling and solving equations much easier and more accurate.

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