Problem 32 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ \frac{63}{k} \text { when } k=9 $$

Step-by-Step Solution

Answer

After substituting \(k\) with 9 and performing the division, the final answer is 7.

Step 1: Understanding the task

In this exercise, we are asked to evaluate the arithmetic expression \(\frac{63}{k}\) for \(k=9\). In other words, substitute \(k\) with 9 and then perform the division.

Step 2: Substituting the value

Substitute the given value of \(k\) into the expression, we have: \(\frac{63}{k} = \frac{63}{9}\)

Step 3: Perform the division

Divide 63 by 9 to evaluate the expression: \(\frac{63}{9} = 7\)

Key Concepts

arithmetic operations

Arithmetic operations are the basic computations used in mathematics. These operations include addition, subtraction, multiplication, and division. In this exercise, division is the primary arithmetic operation used.

When evaluating expressions, understanding arithmetic operations is crucial. They allow you to simplify complex problems into manageable steps. For instance, if you are given an expression like \( \frac{63}{k} \) and asked to evaluate it for a specific value of \(k\), you need to decide which arithmetic operations apply.

• Think of arithmetic operations as the basic rules you follow in math.
• Each step in solving an expression usually involves one or more operations.
• In division, you determine how many times one number fits into another.

Recognize that mastering arithmetic operations is foundational for solving mathematical expressions.

substitution

Substitution is a key concept in algebra that involves replacing a variable with a given value. This is a big part of evaluating expressions like \( \frac{63}{k} \).

When asked to substitute, you literally "swap out" the variable for the number provided. In our example, you substitute \( k = 9 \) into the expression. This changes it to \( \frac{63}{9} \), effectively turning the expression into a numerical equation that can be solved using basic arithmetic.

• Look at the expression and identify the variable to substitute.
• Replace the variable with the value provided.
• After substitution, simplify the numerical expression.

Substitution simplifies the problem and makes it possible to find an exact numerical answer.

division

Division is one of the four fundamental arithmetic operations and it's key to solving the expression \( \frac{63}{9} \). In division, you determine how many times one number (the divisor) fits into another number (the dividend).

Here’s how you perform division in our example:- The dividend is 63.- The divisor is 9.- Ask yourself: how many times does 9 fit into 63?

It fits exactly 7 times, so \( \frac{63}{9} = 7 \). This is the process of division applied to our substituted expression.

• Ensure both the dividend and divisor are clear in your expression.
• Perform the division operation to simplify to one number.
• Understand that division results in a quotient, which is often the final solution to a division problem.

Grasping the concept of division is essential for evaluating expressions and solving problems where you break numbers down into equal parts.