Problem 58 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x-y\) for the given values of \(x\) and \(y.\) $$x=-\frac{15}{16}, y=\frac{5}{16}$$

Step-by-Step Solution

Answer

The evaluated expression \(x-y\) for \(x=-\frac{15}{16}\) and \(y=\frac{5}{16}\) is \(-\frac{5}{4}\).

Step 1: Substitution

Substitute the given \(x\) and \(y\) values into the expression \(x-y\). It will look like this: \(-\frac{15}{16}-\frac{5}{16}\).

Step 2: Perform the subtraction

Now carry out the subtraction operation. Since the denominators are the same, you can just subtract the numerators of the fractions: \(-\frac{15}{16}-\frac{5}{16} = -\frac{20}{16}\).

Step 3: Simplification

The result, \(-\frac{20}{16}\), can be simplified by dividing both the numerator and the denominator by their common factor which is 4. This results into: \(-\frac{5}{4}\).

Key Concepts

Substitute Values

Substituting values in an expression is one of the most fundamental skills in algebra. It's the process of replacing variables with values given in a problem to simplify and solve it.
In the expression provided by our exercise, we have two variables: \(x\) and \(y\). We have specific values to substitute: \(x = -\frac{15}{16}\) and \(y = \frac{5}{16}\).

• Identify the variables within the expression that need substitution.
• Replace each variable with its respective value.
• Ensure all operations and functions of the expression remain intact.

For example, after substituting these given values into the expression \(x-y\), it changes from \(x-y\) to \(-\frac{15}{16} - \frac{5}{16}\). Now, the expression is ready to be simplified or solved.

Like Denominators

When it comes to subtracting fractions, whether you’re using simple arithmetic or solving algebraic expressions, having like denominators simplifies the process greatly. Like denominators mean that the denominators of each fraction in the expression are the same. This allows for a direct subtraction of the numerators.
In our case, both fractions \(-\frac{15}{16}\) and \(\frac{5}{16}\) already have the same denominator of 16. This is beneficial because:

• It eliminates the need to find a common denominator, which can save time and reduce errors.
• It allows you to subtract the numerators directly: simply take \(-15\) minus \(5\), giving you a new numerator of \(-20\).

The expression thus becomes \(-\frac{20}{16}\). This direct subtraction is possible only because the denominators matched. Otherwise, we'd have to first convert the fractions, which could get cumbersome.

Simplify Fractions

Once you have performed operations like addition or subtraction on fractions, the next step is to simplify the result. Simplifying fractions involves reducing a fraction to its lowest terms. This means dividing both the numerator and the denominator by their greatest common divisor (GCD) or simply a common factor.
In our resulting fraction \(-\frac{20}{16}\), we seek the largest number that divides both -20 and 16 without leaving a remainder.

• Here, 4 is the common factor.
• By dividing both the numerator \(-20\) and the denominator 16 by 4, we can simplify \(-\frac{20}{16}\) to \(-\frac{5}{4}\).

Simplifying fractions is crucial as it makes the fraction easier to understand and statistically handle in further mathematical problems or real-world applications. Whether it’s a negative fraction or not, the process remains the same, leading to concise and neat results.