Problem 37 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. Begin your work by rewriting each equation without fractions. $$\frac{x}{3}+\frac{x}{2}=\frac{5}{6}$$

Step-by-Step Solution

Answer

The solution for the equation is x = 1.

Step 1: Rewrite the equation without fractions

The least common denominator (LCD) between 3, 2, and 6 is 6. Multiply each term by 6 to eliminate the fractions: \(6*\frac{x}{3} + 6*\frac{x}{2} = 6*\frac{5}{6}\). This simplifies to \(2x + 3x = 5\). Combining like terms gives \(5x = 5\).

Step 2: Solve for x

Divide the entire equation by 5 to isolate x: \(x = 1\).

Step 3: Check the solution

Substitute x = 1 back into the original equation: \( \frac{1}{3} + \frac{1}{2} = ? \). Simplifying the left side gives \(\frac{5}{6}\), which equals the right side of the equation, verifying that x = 1 is indeed the correct solution.

Key Concepts

Understanding the Least Common Denominator

When working with equations that involve fractions, the first step to simplifying is finding the least common denominator, often abbreviated as LCD. The LCD is the smallest number that each of the denominators in the equation can divide into without leaving a remainder. This concept helps in converting all fractions into whole numbers, making it easier to solve the equation.

For our exercise equation, \(\frac{x}{3} + \frac{x}{2} = \frac{5}{6}\), the denominators are 3, 2, and 6. Here are some steps to find the LCD:

• List the multiples of each denominator: 3 (3, 6, 9...), 2 (2, 4, 6...), and 6 (6, 12, 18...)
• Identify the smallest common multiple, which in this case is 6.
• Use this common denominator to rewrite each term and eliminate the fractions in the equation.

By multiplying all terms by 6, you lift the fractions out, changing the equation to \(2x + 3x = 5\), which is much easier to handle. This approach streamlines solving equations with fractions significantly.

Combining Like Terms

Once you eliminate fractions from an equation, the next step is to simplify it by combining like terms. Like terms are those that have the same variable raised to the same power. Combining them helps to condense the equation further, making it easier to solve.

After rewriting our exercise equation to \(2x + 3x = 5\), we can combine the terms involving \(x\). Here are how you do it:

• Add the coefficients of the like terms together: \(2 + 3 = 5\).
• The equation therefore simplifies to \(5x = 5\).

This simplification process reduces the complexity of the equation, allowing you to focus on isolating the variable to find its value. It's an essential technique for both basic and more advanced algebraic operations.

Checking Solutions

Once you've found the solution to an equation, it's crucial to verify its correctness. This step ensures that no errors occurred during simplification or calculations.

For the equation \(\frac{x}{3} + \frac{x}{2} = \frac{5}{6}\), we found that \(x = 1\). Checking the solution involves substituting this value back into the original equation and ensuring both sides are equal.

• Replace \(x\) with 1: \(\frac{1}{3} + \frac{1}{2} = ?\)
• Find a common denominator for these fractions, which is 6.
• Convert each fraction: \(\frac{1}{3} = \frac{2}{6}\) and \(\frac{1}{2} = \frac{3}{6}\).
• Add them: \(\frac{2}{6} + \frac{3}{6} = \frac{5}{6}\).
• The resulting \(\frac{5}{6}\) matches the right side of the equation, confirming \(x = 1\) is correct.

Checking your solution is a good habit. It's a way to prevent mistakes and ensure the solution truly satisfies the original equation.