Problem 8 - Algebra A Combined Function

Question

Solve each proportion. $$ \frac{6}{11}=\frac{27}{3 x-2} $$

Step-by-Step Solution

Answer

\( x = \frac{103}{6} \)

Step 1: Understand the Problem

The task is to solve the proportion \( \frac{6}{11} = \frac{27}{3x-2} \) for the variable \( x \).

Step 2: Cross Multiplication

Use cross multiplication to eliminate the fractions: Multiply the numerator on one side by the denominator on the other side and vice versa:\[ 6(3x-2) = 27 \times 11 \]

Step 3: Simplify Both Sides

Simplify both sides of the equation:\[ 6(3x-2) = 297 \]Distribute the 6:\[ 18x - 12 = 297 \]

Step 4: Isolate the Variable

Add 12 to both sides to isolate terms with \( x \) on one side:\[ 18x = 297 + 12 \]Simplify:\[ 18x = 309 \]

Step 5: Solve for x

Divide both sides by 18 to solve for \( x \):\[ x = \frac{309}{18} \]Simplify the fraction:\[ x = \frac{103}{6} \]

Key Concepts

Cross Multiplication

Cross multiplication is a handy technique used to solve proportions, like in our example, \( \frac{6}{11} = \frac{27}{3x-2} \). It works by eliminating fractions and directly equating the products of the diagonally opposite terms in the equation. Here's how to do it:

• Multiply the numerator on the left side by the denominator on the right side: \( 6(3x-2) \).
• Multiply the numerator on the right side by the denominator on the left side: \( 27 \times 11 \).
• Set the two products equal to each other: \( 6(3x-2) = 27 \times 11 \).

By applying cross multiplication, the fractions are eliminated, making it easier to solve for the unknown variable \( x \) by working with simpler algebraic expressions. This method is beneficial as it transforms a complex-looking fraction equation into a straightforward algebraic problem.

Algebraic Equations

An algebraic equation features unknown variables and can be solved to find these variables' values. After using cross multiplication on the proportion \( \frac{6}{11} = \frac{27}{3x-2} \), you end up with an algebraic equation: \[6(3x-2) = 297.\]Here's a step-by-step approach to solving it:

• Distribute the 6 on the left side: \( 18x - 12 \).
• Re-write the equation as \( 18x - 12 = 297 \).
• Isolate the term with \( x \) by adding 12 to both sides: \( 18x = 309 \).
• Simplify the equation to solve for \( x \): \( x = \frac{309}{18} \).

Working with algebraic equations involves performing operations that simplify the equation step by step until the variable is isolated, allowing you to solve for its value. Understanding how to manipulate algebraic expressions is key to solving these types of problems.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that helps us make numbers more convenient and manageable. After finding \( x = \frac{309}{18} \) in our example from the cross multiplication and algebraic steps, the next task is to simplify this fraction.

• Look for the greatest common divisor (GCD) of the numerator and the denominator to simplify the fraction effectively.
• In our case, \( 309 \) and \( 18 \) have a GCD of 3.
• Divide both the numerator and denominator by their GCD: \( x = \frac{309}{18} = \frac{309 \div 3}{18 \div 3} = \frac{103}{6} \).

Simplifying the fraction involves reducing it to its lowest terms by dividing both the top and bottom numbers by their common factors. This process results in a neater answer that is easier to interpret and use in further calculations.