Problem 8 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{9}{4 x}+\frac{1}{3}=\frac{5}{2 x} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{4}\).

Step 1: Identify the Least Common Denominator (LCD)

First, observe the denominators of the fractions in the equation: \(4x\), \(3\), and \(2x\). To combine these fractions, we need a common denominator. The LCD here would be the least common multiple of these denominators. The LCM of \(4x\), \(3\), and \(2x\) is \(12x\).

Step 2: Clear the Fractions

Multiply every term in the equation by \(12x\) to eliminate the fractions:\[12x \left( \frac{9}{4x} \right) + 12x \left( \frac{1}{3} \right) = 12x \left( \frac{5}{2x} \right)\]Simplifying each term:- \(12x \cdot \frac{9}{4x} = 27\) (because \(12x/4x = 3\), and \(3 \cdot 9 = 27\))- \(12x \cdot \frac{1}{3} = 4x\)- \(12x \cdot \frac{5}{2x} = 30\) (because \(12x/2x = 6\), and \(6 \cdot 5 = 30\))Resulting in: \[27 + 4x = 30\]

Step 3: Solve for x

Now, solve the linear equation \(27 + 4x = 30\):- Subtract \(27\) from both sides:\[4x = 3\]- Divide both sides by \(4\) to solve for \(x\):\[x = \frac{3}{4}\]

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(x = \frac{3}{4}\) back into the original equation and check if both sides equal:- Substitute into \(\frac{9}{4(\frac{3}{4})} + \frac{1}{3} = \frac{5}{2(\frac{3}{4})}\)- Simplify: \(\frac{9}{3} + \frac{1}{3} = \frac{5}{1.5}\)- Which reduces to: \(3 + \frac{1}{3} = \frac{10}{3}\)- Thus, both sides simplify to the same value confirming \(x = \frac{3}{4}\) is correct.

Key Concepts

Understanding the Least Common Denominator (LCD)

When you encounter a rational equation, like the one provided, the first challenge is managing different denominators. The least common denominator (LCD) is crucial in combining fractions. It allows us to clear the fractions and simplify the equation. To find the LCD, identify the smallest number that is a multiple of all denominators in the equation.

In our example, the denominators are \(4x\), \(3\), and \(2x\). Since \(x\) is a common factor in \(4x\) and \(2x\), the task is to find the least common multiple of the coefficients, \(2\), \(3\), and \(4\).
Consider these steps:

• List multiples of each coefficient until a common multiple is found.
• The smallest number that appears in each list is the LCD.

In this case, that number is \(12\), therefore, our LCD is \(12x\). With the least common denominator identified, we can eliminate the fractions by multiplying through the equation.

Solving Linear Equations

Once the fractions are cleared, you are usually left with a linear equation. A linear equation is simply an equation where the highest power of the variable is 1. These equations are straightforward to solve and involve basic algebraic steps.

For example, after clearing fractions in our given exercise, we have the equation \(27 + 4x = 30\). Solving it involves:

• Moving constant terms to one side. Here, subtract \(27\) from both sides to isolate terms involving \(x\) on one side: \(4x = 3\).
• Dividing each side by the coefficient of \(x\). For this example, divide by \(4\) to find its value as \(x = \frac{3}{4}\).

This approach is standard for solving linear equations. Be sure to always verify your answer by substituting back into the original equation.

Utilizing Cross-Multiplication

Cross-multiplication is an important step when dealing with equations involving fractions. Although it's not directly used in our problem, understanding it helps in recognizing situations where it becomes relevant in solving rational equations.

Cross-multiplication is typically applied when two fractions are equal to each other, like \(\frac{a}{b} = \frac{c}{d}\). In such cases, cross-multiplying involves multiplying the outer terms and setting them equal to the product of the inner terms:

• Multiply \(a\) and \(d\), then Multiply \(b\) and \(c\).
• Equate these products: \(a \cdot d = b \cdot c\).

Using cross-multiplication simplifies solving the equation by transforming it into a simpler format. Understanding when to use cross-multiplication can make handling equations involving fractions much easier.