Problem 1 - Elementary Algebra

Question

\(\frac{x}{2}+\frac{x}{3}=10\)

Step-by-Step Solution

Answer

The solution is \(x = 12\).

Step 1: Find a Common Denominator

First, we need to add the fractions \(\frac{x}{2}\) and \(\frac{x}{3}\). To do this, we need a common denominator. The least common multiple of 2 and 3 is 6. Hence, rewrite each fraction with a denominator of 6.

Step 2: Rewrite Fractions

Convert the terms to have a common denominator: \( \frac{x}{2} = \frac{3x}{6} \) and \( \frac{x}{3} = \frac{2x}{6} \). Now, the equation is \( \frac{3x}{6} + \frac{2x}{6} = 10 \).

Step 3: Combine Fractions

Combine the fractions on the left side: \[ \frac{3x}{6} + \frac{2x}{6} = \frac{5x}{6} \]. The equation now is \( \frac{5x}{6} = 10 \).

Step 4: Clear the Fraction by Multiplying

To eliminate the fraction, multiply both sides of the equation by 6: \( 6 \times \frac{5x}{6} = 6 \times 10 \). This simplifies to \( 5x = 60 \).

Step 5: Solve for \(x\)

To isolate \(x\), divide both sides of the equation by 5: \( \frac{5x}{5} = \frac{60}{5} \). This results in \( x = 12 \).

Key Concepts

Fractions in Equations

When solving equations that include fractions, it's important to understand how to handle and manipulate them. Fractions often appear in algebraic equations, and knowing how to work with them is critical for solving these equations effectively.

• Each fraction consists of a numerator (the top value) and a denominator (the bottom value).
• In algebraic terms, the fractions may include variables like "x" which must be carefully handled during equation operations.
• In our example problem, the fractions \( \frac{x}{2} \) and \( \frac{x}{3} \) need to be combined.

When combining fractions in equations, typically the goal is to find an equivalent expression where the fractions are combined into a single term. Understanding the role and behavior of fractions within the equation can make it easier to visualize and perform the necessary manipulations.

Common Denominator

Finding a common denominator is a key step when dealing with fractions in equations. The common denominator allows the addition or subtraction of two fractions by making their denominators the same.

• The least common multiple (LCM) of the denominators is used to determine the lowest common denominator possible.
• For example, the LCM of 2 and 3 is 6, which we use to rewrite the fractions \( \frac{x}{2} \) and \( \frac{x}{3} \) as equivalent fractions \( \frac{3x}{6} \) and \( \frac{2x}{6} \) respectively.

This enables us to combine the fractions because they share a common denominator. It simplifies the process and makes it much easier to proceed to solve the equation effectively. Without the common denominator, it would be impossible to directly add or subtract these fractions.

Simplifying Equations

Simplifying equations is the process of making them easier to work with by reducing complexity. After ensuring the fractions have a common denominator, the next step is to simplify the equation further.

• In the example provided, once we have \( \frac{3x}{6} + \frac{2x}{6} \), we can combine them to get \( \frac{5x}{6} \).
• The goal is to isolate the unknown variable (in this case, \( x \)) to determine its value.

To simplify even further, you clear the fractions by multiplying both sides of the equation by the common denominator (6). This step eliminates the fraction, resulting in a straightforward algebraic equation \[5x = 60\]. Finally, dividing both sides by the constant term, which is 5, isolates \( x \), giving the solution \( x = 12 \). Simplifying equations isn't just about removing fractions—it’s about making the equation as simple as possible to solve for the unknown values. This methodical approach ensures accuracy throughout the solving process.