Problem 18 - Elementary and Intermediate Algebra

Question

The Try Exercises for examples are indicated by a shaded block on the exercise number. Answers to these exercises appear at the end of the exercise set as well as at the back of the book. $$ \frac{c+d}{7}, \text { for } c=15 \text { and } d=20 $$

Step-by-Step Solution

Answer

5

Step 1 - Write down the given expression

The given expression is \( \frac{c+d}{7} \).

Step 2 - Substitute the values of c and d

We know that \( c = 15 \) and \( d = 20 \). Substitute these values into the expression: \( \frac{15 + 20}{7} \).

Step 3 - Add the values of c and d

First, add the values: \( 15 + 20 = 35 \). This gives us \( \frac{35}{7} \).

Step 4 - Simplify the fraction

Divide 35 by 7: \( \frac{35}{7} = 5 \).

Key Concepts

algebraic expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They help us represent real-life situations using symbols. In the given exercise, \( \frac{c+d}{7} \) is an algebraic expression. Here, \(c\) and \(d\) are variables that can take different values. By identifying and understanding these expressions, we can solve problems by substituting specific numbers for the variables.

substitution method

The substitution method involves replacing the variables in an expression with their given values. This is a crucial technique in algebra. In our example, we substitute \( c = 15 \) and \( d = 20 \) into the expression \( \frac{c+d}{7} \). After substitution, it becomes \( \frac{15 + 20}{7} \). By performing this step, we transform a general expression into a specific arithmetic problem that can be solved immediately.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). In the exercise, after substituting and adding the values, we have \( \frac{35}{7} \). Both 35 and 7 are divisible by 7, the GCD of the two numbers. By performing the division, we get \( 5 \), which is the simplest form of the fraction. Simplifying fractions makes them easier to understand and work with.

arithmetic operations

Arithmetic operations include addition, subtraction, multiplication, and division. These basic operations are the foundation of mathematics. Let’s break down the exercise:

• First, we performed addition (\(15 + 20 = 35\)).
• Next, we performed division (\( \frac{35}{7} \)).

Understanding and mastering these operations are essential for solving various mathematical problems. In this exercise, these operations help us to progress from an algebraic expression to a simplified numerical result.