Problem 234 - Intermediate Algebra

Question

In the following exercises, evaluate. \(\frac{r-s}{r+s}\) when $$ r=10, s=-5 $$

Step-by-Step Solution

Answer

3

Step 1: Substitute the given values

First, substitute the given values of the variables into the expression. Given that \( r = 10 \) and \( s = -5 \), substitute these values into the expression \( \frac{r-s}{r+s} \). This results in: \( \frac{10-(-5)}{10+(-5)} \).

Step 2: Simplify the expression

Simplify the expression inside the numerator and the denominator. \( 10 - (-5) = 10 + 5 = 15 \) \( 10 + (-5) = 10 - 5 = 5 \) Therefore, the expression simplifies to: \( \frac{15}{5} \).

Step 3: Divide the simplified terms

Finally, divide the simplified numerator by the simplified denominator. \( \frac{15}{5} = 3 \).

Key Concepts

Substitution in Algebra

Substitution in algebra is key to solving various types of problems. It involves replacing variables in an expression with their given values. This allows us to convert a general formula into a specific calculation.
In our exercise, we are given specific values for \( r \) and \( s \): \( r = 10 \) and \( s = -5 \). By substituting these values into the rational expression \( \frac{r-s}{r+s} \), we get:

\( \frac{10 - (-5)}{10 + (-5)} \).

Always pay attention to the signs while performing substitution. This step is crucial for further simplifying the expression accurately.

Simplifying Algebraic Expressions

After substitution, your next task is to simplify the algebraic expression. Simplifying involves performing arithmetic operations to combine like terms and reduce the expression to its simplest form. Here, you can follow a sequence of steps to simplify the numerator and the denominator separately:

• Numerator: \( 10 - (-5) \) results in \( 10 + 5 = 15 \)
• Denominator: \( 10 + (-5) \) simplifies to \( 10 - 5 = 5 \)

Once the numerator and denominator are simplified, the rational expression becomes \( \frac{15}{5} \). Simplifying each part makes the expression more manageable and prepares it for the final operation.

Division in Algebra

Finally, the simplified terms are divided to find the final value of the expression. Division in algebra is straightforward -- the numerator is divided by the denominator. From our simplified expression \( \frac{15}{5} \), we perform the division:

\( 15 \, \text{divided by} \, 5 = 3 \).

This step completes our evaluation of the given rational expression. It is crucial to double-check your simplification and arithmetic to ensure accuracy, as mistakes in early steps can lead to incorrect final results.