Problem 84 - Algebra 2

Question

Solve each equation. \(x+\frac{5}{8}=-\frac{5}{6}\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{35}{24}\).

Step 1: Isolate Variable x

To solve for \(x\), we need to subtract \(\frac{5}{8}\) from both sides of the equation to isolate \(x\). This gives us: \[ x = -\frac{5}{6} - \frac{5}{8} \]

Step 2: Find a Common Denominator

To subtract the fractions \(-\frac{5}{6}\) and \(\frac{5}{8}\), we need a common denominator. The least common multiple of 6 and 8 is 24. Convert each fraction: \[ -\frac{5}{6} = -\frac{20}{24}, \quad \frac{5}{8} = \frac{15}{24} \]

Step 3: Perform the Subtraction

Subtract the fractions using their common denominator: \[ x = -\frac{20}{24} - \frac{15}{24} = -\left(\frac{20 + 15}{24}\right) = -\frac{35}{24} \]

Step 4: Simplify the Result if Possible

The fraction \(-\frac{35}{24}\) cannot be simplified further since 35 and 24 have no common factors other than 1. Therefore, \[ x = -\frac{35}{24} \] is the solution.

Key Concepts

Fraction Subtraction

Subtracting fractions might seem tricky at first, but once you get the hang of it, it's straightforward. To subtract two fractions, like in the equation \(x = -\frac{5}{6} - \frac{5}{8}\), you need to ensure that the fractions have a common denominator. Once the denominators match, subtraction becomes simple. You just subtract the numerators and keep the common denominator. For instance:

• Subtract \(-\frac{20}{24}\) and \(\frac{15}{24}\) results in \(-\frac{35}{24}\).

This straightforward approach turns any timid fraction subtraction into child's play!

Common Denominator

Finding a common denominator is a crucial step when adding or subtracting fractions. It helps us to combine the fractions seamlessly. Consider the fractions \(\frac{5}{6}\) and \(\frac{5}{8}\) in our example. They have different denominators, 6 and 8, respectively. To find a common base for these denominators, look for the least common multiple (LCM).

• The LCM of 6 and 8 is 24. This is because 24 is the smallest number that both 6 and 8 can divide evenly.

Once the common denominator is found, adjust each fraction accordingly, making them equivalent fractions with the same base. This adjustment allows us to perform operations on the numerators easily.

Isolate Variable

Isolating the variable is a fundamental technique in solving equations. In the given equation \(x + \frac{5}{8} = -\frac{5}{6}\), our goal was to get \(x\) by itself on one side of the equation. We achieve this by removing the fraction \(\frac{5}{8}\) from the left side. To do so:

• Simply subtract \(\frac{5}{8}\) from both sides.

This keeps the equation balanced and simplifies the path to finding \(x\). Remember, what you do to one side, do to the other. This principle keeps both sides of the equation equal and preserves the integrity of the original equation.

Simplifying Fractions

Simplifying fractions is about making fractions as clear and concise as possible. If a fraction has a numerator and a denominator with common factors except 1, they can be reduced to a simpler form by dividing both by their greatest common divisor (GCD). For example:

• If you have \(\frac{8}{12}\), both 8 and 12 can be divided by 4, which is their GCD. Simplified, it becomes \(\frac{2}{3}\).

However, in our solution for \(x = -\frac{35}{24}\), 35 and 24 share no common factors other than 1. Thus, this fraction is already in its simplest form. It's essential to check for simplification to ensure your final answer looks neat and is easily understandable.