Problem 10 - Intermediate Algebra

Question

a. When solving \(\frac{x+1}{3}-\frac{2}{15}=\frac{x-1}{5},\) why would we multiply both sides by \(15 ?\) b. When solving \(1.45 x-0.5(1-x)=0.7 x,\) why would we multiply both sides by \(100 ?\)

Step-by-Step Solution

Answer

Multiply by 15 to eliminate fractions. Multiply by 100 to handle decimals.

Step 1: Identify the Given Equation

The given equation is \( \frac{x+1}{3} - \frac{2}{15} = \frac{x-1}{5} \).

Step 2: Identify the Denominators

Observe that the denominators in the expression are 3, 15, and 5.

Step 3: Determine the Least Common Multiple (LCM) for Denominators

The LCM of the denominators 3, 15, and 5 is 15 because 15 is the smallest number that 3, 5, and 15 all divide into evenly.

Step 4: Multiply Both Sides by the LCM

To eliminate the fractions, multiply both sides by 15. This will cancel out the denominators in the equation making it easier to solve.

Key Concepts

Solving Linear Equations

When tackling linear equations, the primary goal is to isolate the variable—in most cases, denoted as \( x \). The objective is to manipulate the equation until \( x \) stands alone on one side of the equation. This involves performing operations that simplify or eliminate any terms obstructing this goal. These operations include addition, subtraction, multiplication, and division.

In the context of linear equations with fractions, the usual barrier to straightforward solving is the presence of multiple fractional terms. Thus, simplification techniques, such as aiming to have whole numbers rather than fractions, are often employed to make solving easier.

• Isolate terms: Rearrange the equation so all terms containing the variable are on one side, and all constant terms are on the other side.
• Combine like terms: Simplify the equation by combining similar terms on each side.
• Perform inverse operations: Use addition and subtraction to solve for the variable.

With practice, recognizing these operations becomes intuitive, allowing one to solve linear equations more efficiently.

Least Common Multiple

The least common multiple (LCM) is an essential concept when dealing with fractions in equations. It refers to the smallest number that is a multiple of all denominators in the equation. Identifying the LCM is crucial for eliminating fractions, as multiplying each term of the equation by this number clears the fractions away by canceling out denominators.

To find the LCM:

• List the prime factors of each denominator.
• Choose the highest power of each prime number that appears across the lists.
• Multiply these chosen factors together to get the LCM.

In the example of solving \( \frac{x+1}{3} - \frac{2}{15} = \frac{x-1}{5} \), the LCM of 3, 15, and 5 is 15. By multiplying the entire equation by 15, each term loses its denominator, making the equation simpler and more straightforward to solve. This step is pivotal as it transforms complex fractional equations into simpler, linear ones with whole numbers.

Eliminating Fractions

Fractions can make equations more complex, and eliminating them often simplifies the solving process. Multiplying the entire equation by a common factor—such as the LCM of the denominators—turns fractions into whole numbers by effectively distributing the LCM across each term.

This process involves a few organized steps:

• Determine the LCM: As discussed, identify the least common multiple of all denominators in the equation.
• Multiply each term: Apply the LCM to every term in the equation, including both sides.
• Simplify the equation: Cancel the denominators with their respective numerators, yielding a simplified expression with whole numbers.

By doing so in the given example, the fractions \( \frac{x+1}{3} \), \( \frac{2}{15} \), and \( \frac{x-1}{5} \) are cleared by multiplying through with 15, simplifying the path to solving for \( x \). This method integrates with solving linear equations to remove the complication of fractions, making the solving process significantly more manageable.