Problem 17 - Prealgebra

Question

Solve each equation. Check your solution. $$m+\frac{7}{12}=-\frac{5}{18}$$

Step-by-Step Solution

Answer

The solution is \(m = -\frac{31}{36}\).

Step 1: Isolate the Variable

We're given the equation \(m + \frac{7}{12} = -\frac{5}{18}\). To isolate \(m\), we need to subtract \(\frac{7}{12}\) from both sides of the equation: \[ m = -\frac{5}{18} - \frac{7}{12} \]

Step 2: Find a Common Denominator

To subtract fractions, they must have a common denominator. The denominators are 18 and 12. The least common multiple of 18 and 12 is 36. Convert \(-\frac{5}{18}\) and \(\frac{7}{12}\) to have a denominator of 36: \[ -\frac{5}{18} = -\frac{5 \times 2}{18 \times 2} = -\frac{10}{36} \]\[ \frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36} \]

Step 3: Subtract the Fractions

Now subtract the two fractions: \[ m = -\frac{10}{36} - \frac{21}{36} = -\frac{10 + 21}{36} = -\frac{31}{36} \]

Step 4: Check the Solution

Substitute \( m = -\frac{31}{36} \) back into the original equation to ensure it satisfies it: \[ -\frac{31}{36} + \frac{7}{12} = -\frac{5}{18} \]Convert \(\frac{7}{12}\) to \(\frac{21}{36}\) as in Step 2:\[ -\frac{31}{36} + \frac{21}{36} = -\frac{10}{36} = -\frac{5}{18} \]Since both sides equal \(-\frac{5}{18}\), the solution is verified.

Key Concepts

Fractions

Fractions are a way of representing parts of a whole. They consist of two parts: a numerator and a denominator. The numerator is the top part of the fraction and indicates how many parts of the whole you have. The denominator is the lower part and tells you how many parts the whole is divided into.
When dealing with fractions in equations, it's important to work carefully to ensure accuracy. You might be asked to add, subtract, multiply, or divide fractions, which often involves other steps, such as finding a common denominator or simplifying the result. Understanding fractions is key to solving equations that contain them.

Common Denominator

To subtract or add fractions, they must have the same denominator. This is called a common denominator. The denominator represents the total number of equal parts a whole is divided into. Without a common denominator, you cannot directly perform arithmetic operations on fractions.
Finding a common denominator involves identifying the least common multiple (LCM) of the denominators. In our example, we had the fractions \(-\frac{5}{18}\) and \(\frac{7}{12}\). Their denominators were 18 and 12. By determining the LCM, we found that 36 was the smallest number that both 18 and 12 could divide into equally.
Once a common denominator is found, convert each fraction. Multiply both the numerator and denominator of each fraction by the required factor to achieve the common denominator. This step allows you to easily add or subtract fractions.

Subtracting Fractions

Once fractions have a common denominator, subtracting them becomes straightforward. All you need to do is subtract the numerators while keeping the denominator the same.
For example, consider the fractions \(-\frac{10}{36}\) and \(\frac{21}{36}\). They have a common denominator of 36. Subtract the numerators as follows:

• Start with the first numerator: -10.
• Subtract the second numerator from it: -10 - 21.
• The result is: -31.

The subtraction operation results in \(-\frac{31}{36}\). This means that once the numerators are correctly subtracted, you have your new fraction.

Isolating Variables

Isolating a variable means rearranging an equation to get the variable by itself on one side of the equation. This process allows you to solve for the unknown variable.
In equations involving fractions, isolation often requires additional steps like finding common denominators before adding or subtracting them. In our exercise, the equation was \( m + \frac{7}{12} = -\frac{5}{18} \). To isolate \( m \), we needed to subtract \( \frac{7}{12} \) from each side of the equation:

• Move \( \frac{7}{12} \) across the equals sign by subtracting it, leading to: \( m = -\frac{5}{18} - \frac{7}{12} \).
• Find a common denominator and subtract the fractions to solve for \( m \).

This process is fundamental in solving algebraic equations, especially when variables are part of fractional expressions.