Problem 39 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \frac{5}{7}-\frac{4}{7}-\left(-\frac{6}{7}\right) $$

Step-by-Step Solution

Answer

\(\frac{7}{7}=1\).

Step 1: Identify the Operations

Firstly, recognize the subtraction operations and a subtraction of negative operation in the exercise \(\frac{5}{7}-\frac{4}{7}-\left(-\frac{6}{7}\right)\).

Step 2: Convert Subtraction of Negative to Addition

The subtraction of a negative number is the same as adding a positive. So the expression becomes \(\frac{5}{7}-\frac{4}{7}+\frac{6}{7}\).

Step 3: Perform the Operations

Carry out the operations by adding and subtracting the fractions directly because they have the same denominator. The resulting value will be \(\frac{5}{7}-\frac{4}{7}+\frac{6}{7}=\frac{7}{7}\)

Key Concepts

Fraction Subtraction

Understanding how to subtract fractions is a fundamental skill required to tackle many mathematical problems. Working with fractions using their denominators and numerators can initially be daunting, but when you grasp the concept, it becomes second nature. For the subtraction of fractions with like denominators, it's relatively straightforward: keep the denominator the same and subtract the numerators. Take an example of two fractions, \(\frac{a}{c} - \frac{b}{c}\), the subtraction process involves taking away the value of the second numerator from the first, resulting in \(\frac{a-b}{c}\).

Let's apply this to our exercise \( \frac{5}{7}-\frac{4}{7} \) by simply subtracting 4 from 5, which gives us \(\frac{5-4}{7} = \frac{1}{7}\). It is essential to remember that this method only works when the denominators are equal; otherwise, you would need to find a common denominator before subtracting.

Subtracting Negative Fractions

The concept of subtracting negative fractions is connected to understanding the rules of operation with negative numbers. When you encounter a negative sign in front of a fraction, it's vital to recognize that subtracting a negative value is equivalent to adding its positive counterpart. Therefore, the expression \( \frac{5}{7} - \left(-\frac{6}{7}\right) \) becomes \( \frac{5}{7} + \frac{6}{7} \) when the negative sign is addressed correctly.

In the context of our exercise, after recognzing the subtraction of the negative, we replace the \( -\left(-\frac{6}{7}\right) \) with \( +\frac{6}{7} \) as per Step 2 in the provided solution. This change simplifies the equation and clears the path for straightforward addition, which further simplifies the process of solving the expression.

Adding Fractions with Like Denominators

Adding fractions that share the same denominator is an easy task. The rule here is quite intuitive: keep the common denominator the same while adding the numerators together. For example, when combining \(\frac{x}{z} + \frac{y}{z}\), you obtain \(\frac{x+y}{z}\). This operation is made simple by the fact that you're not changing the parts of each fraction that represent the whole; you're simply combining the parts that you have.

In the context of our exercise, \(\frac{5}{7}-\frac{4}{7}+\frac{6}{7}\), we've combined fractions by adding the numerators (after properly dealing with the negative fraction), leading to \(\frac{5-4+6}{7}\), which simplifies to \(\frac{7}{7}\). In this case, the result is a whole number, 1, because the numerator and denominator are equal. Remember, when adding fractions, it's crucial to only combine like terms—that is, terms with the same denominators.