Problem 49 - Algebra 1: Concepts and Skills

Question

Find the sum. $$\frac{1}{7}+(-2)+\left(-\frac{5}{7}\right)$$

Step-by-Step Solution

Answer

The sum is \(-2 \frac{4}{7}\).

Step 1: Identify the Terms

The terms that need to be added together are \( \frac{1}{7} \), \(-2\), and \(-\frac{5}{7}\).

Step 2: Combine Like terms

Start by adding the fractions together: \( \frac{1}{7} - \frac{5}{7} = - \frac{4}{7} \). The result is a negative fraction.

Step 3: Add the result with the integer

Now, add the result from step 2 with the integer –2 : \( - \frac{4}{7} + (-2) = - \frac{4}{7} - 2 = -\frac{4+14}{7} = -\frac{18}{7} \). \(-\frac{18}{7}\) simplifies to \(-2 \frac{4}{7}\).

Key Concepts

Understanding Fractions

A fraction represents a part of a whole. It consists of a numerator and a denominator. The numerator is the top number and shows how many parts we have. The denominator is the bottom number and shows how many equal parts the whole is divided into. For instance, in the fraction \( \frac{1}{7} \), 1 is the numerator and 7 is the denominator. This tells us that we have 1 part out of 7 equal parts.
Working with fractions often requires having a common denominator, especially when adding or subtracting them. If the denominators are the same, you can directly add or subtract the numerators. If not, you'll need to find a common denominator first. In the given exercise, both fractions \( \frac{1}{7} \) and \(-\frac{5}{7} \) have the same denominator, making it straightforward to add them.

Adding Integers

Integers include both positive and negative whole numbers, such as -2 or 3. When adding integers, consider the signs of the numbers you are combining. Like signs mean you can add the numbers and keep the sign. Opposite signs mean you find the difference between the numbers and use the sign of the bigger number. For example, adding \(-2\) with \(-\frac{4}{7}\) involves adding a negative fraction with a negative whole number, resulting in a larger negative value.
When combining these types of numbers, it is essential to remember to keep track of the signs. In mathematics, the order of addition doesn't change the sum, but ensuring we have the correct signs and understanding how they influence each other is crucial for getting the right answer.

Simplifying Rational Expressions

Rational expressions are fractions with polynomials as the numerator and/or the denominator. Although the given exercise deals with numerical fractions, the principles of simplifying rational expressions apply. Simplifying involves reducing the expression to its simplest form. This can mean combining like terms or breaking down into a more understandable form. For \(-\frac{18}{7}\), we can see it as a mixed number: \(-2\frac{4}{7}\). This involves dividing the numerator by the denominator to get a whole number and a new fraction.
When simplifying, always check if the fraction can be reduced. However, since 18 and 7 share no common factors besides 1, \(-2\frac{4}{7}\) is the simplest form. Simplifying expressions helps in understanding the results clearly and aids in comparing or further manipulating those numbers or expressions.