Problem 21 - 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. $$ \frac{9 m}{q}, \text { for } m=6 \text { and } q=18 $$

Step-by-Step Solution

Answer

The value is 3.

Step 1: Understand the Formula

The formula given is \( \frac{9m}{q} \).

Step 2: Substitute the Given Values

Substitute \( m = 6 \) and \( q = 18 \) into the formula: \( \frac{9 \times 6}{18} \).

Step 3: Perform the Multiplication

Multiply the numerator: \( 9 \times 6 = 54 \), so the expression becomes \( \frac{54}{18} \).

Step 4: Simplify the Fraction

Divide 54 by 18 to simplify: \( \frac{54}{18} = 3 \).

Key Concepts

algebraic formulas

An algebraic formula is a mathematical expression that connects variables and constants using arithmetic operations like addition, subtraction, multiplication, and division.
These formulas are very powerful tools because they allow us to represent complex situations and solve problems in a simplified form.
In our example with the formula \(\frac{9m}{q}\), we use algebra to relate variables m and q to get a specific outcome.

When dealing with algebraic formulas, it's important to understand the role of each variable and constant involved.

• Constansts are values that don't change. In the formula \(\frac{9m}{q}\), the number 9 is a constant.
• Variables are symbols that represent unknown or changeable values. Here, m and q are variables.

substitution in algebra

Substitution in algebra involves replacing variables with known values to simplify and solve expressions.
It’s like plugging in the numbers to see the solution in action.
In the problem, we are given the values for m and q: \ m = 6 \ and \ q = 18 \.

The steps to substitute these values are straightforward:

• First, identify the values given for each variable.
• Next, replace the variables in the formula with these values. For our formula \(\frac{9m}{q}\), we plug in m = 6 and q = 18, getting \(\frac{9 \times 6}{18}\).

This technique makes equations more manageable and helps us find precise solutions.
It’s a fundamental skill in algebra that you'll use repeatedly.

multiplication and division in algebra

Multiplication and division are core operations in algebra that you will use almost daily.
In the given problem, both operations come into play to reach the solution.

Step 3 of the problem tells us to perform multiplication:

• Here, multiply 9 by 6 to get 54. The multiplication step makes the problem simpler by eliminating one variable.

After multiplication, we move to division:

• To simplify \(\frac{54}{18}\), we divide 54 by 18. This step reduces the fraction to its simplest form, resulting in 3.

Understanding how to properly multiply and divide in algebra will make solving equations much more straightforward.
Remember, when simplifying fractions, always look for the greatest common divisor, as it makes the simplification process easier and quicker.