Problem 61 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ \frac{3}{4} x-1+\frac{1}{2} x=\frac{5}{12} x+\frac{1}{6} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{7}{5} \).

Step 1: Combine like terms on the left

Start by combining the like terms on the left side of the equation. The expression is \( \frac{3}{4}x - 1 + \frac{1}{2}x \). The like terms are \( \frac{3}{4}x \) and \( \frac{1}{2}x \). To add these, find a common denominator, which is 4. Convert \( \frac{1}{2} \) to \( \frac{2}{4} \) so you get: \( \frac{3}{4}x + \frac{2}{4}x = \frac{5}{4}x \). Now the left side is \( \frac{5}{4}x - 1 \).

Step 2: Rearrange the equation

Rewrite the equation with the combined terms from Step 1: \( \frac{5}{4}x - 1 = \frac{5}{12}x + \frac{1}{6} \).

Step 3: Eliminate fractions by finding a common denominator

To simplify solving the equation, eliminate fractions by using the least common denominator (LCD) of 12. Multiply every term in the equation by 12: \( 12 \times \frac{5}{4}x - 12 \times 1 = 12 \times \frac{5}{12}x + 12 \times \frac{1}{6} \). Simplify to get \( 15x - 12 = 5x + 2 \).

Step 4: Isolate variable terms on one side

Subtract \(5x\) from both sides to get the variable terms on one side: \( 15x - 5x - 12 = 5x - 5x + 2 \). This simplifies to \( 10x - 12 = 2 \).

Step 5: Solve for x

Add 12 to both sides to isolate terms involving \(x\): \( 10x = 14 \). Divide both sides by 10 to solve for \(x\): \( x = \frac{14}{10} \). Simplify \( \frac{14}{10} \) by dividing both the numerator and the denominator by 2, which results in \( x = \frac{7}{5} \).

Key Concepts

Combining Like Terms

In algebra, combining like terms is crucial for simplifying expressions. Like terms are terms that have the same variable raised to the same power. In this case, we look at terms with the variable \(x\). For example, in our problem, we have \(\frac{3}{4}x\) and \(\frac{1}{2}x\) on the left side of the equation.

To combine these, we need to ensure they have a common denominator. The coefficients of these terms are fractions, specifically \(\frac{3}{4}\) and \(\frac{1}{2}\). We find that the common denominator is 4. So \(\frac{1}{2}\) becomes \(\frac{2}{4}\). This allows us to add them as fractions:

• \(\frac{3}{4}x + \frac{2}{4}x = \frac{5}{4}x\).

The term \(x\) remains unchanged, as it is just a factor multiplied by the summed coefficients.

By combining like terms, we reduce the complexity of the equation and make it easier to solve further along the process.

Fraction Operations

Fraction operations, particularly addition and subtraction, often appear daunting, but they become straightforward once you understand the steps. When dealing with fractions, remember to:

• Identify a common denominator.
• Convert each fraction to an equivalent form using that common denominator.
• Perform the necessary arithmetic operation (addition or subtraction).

In the given exercise, the common denominator for combining the terms was 4, as shown in the previous section. For other operations in the solution, we used an LCD of 12 to eliminate fractions.

Always ensure you apply the arithmetic operation only to the numerators, while the denominator remains unchanged until simplification is possible. Practicing these steps will help you manage fractions effectively.

Eliminating Fractions

Eliminating fractions can greatly simplify solving equations. This is usually done by determining an appropriate least common denominator (LCD) for the entire equation and then multiplying each term by that LCD.

In our exercise, once we combined like terms, the equation was:

• \(\frac{5}{4}x - 1 = \frac{5}{12}x + \frac{1}{6}\)

Here, the LCD is 12. By multiplying each term by 12, we get:

• \(12 \times \frac{5}{4}x\) becomes \(15x\)
• \(-12 \times 1\) becomes \(-12\)
• \(12 \times \frac{5}{12}x\) becomes \(5x\)
• \(12 \times \frac{1}{6}\) becomes \(2\)

This eliminates all the fractions, simplifying our equation to \(15x - 12 = 5x + 2\). This makes the equation easier to manipulate and solve for the variable \(x\), as the troublesome fractional components have been removed.