Problem 65 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \begin{equation} \frac{9}{p} \text { when } p=3 \end{equation}

Step-by-Step Solution

Answer

The value of the expression \(\frac{9}{p}\) when \(p=3\) is 3.

Step 1: Understand the exercise

In this exercise, there is an expression \(\frac{9}{p}\) and the variable \(p\) in the expression is given a value, which is 3. The task is to substitute \(p\) with 3 and then to evaluate the expression. The final goal is to simplify the given mathematical formula by replacing the variable with its given value.

Step 2: Substitute the value

Now, replace the \(p\) in the expression with its given value. After substitution, the expression becomes \(\frac{9}{3}\). The next step is to evaluate the expression.

Step 3: Evaluate the expression

Perform the operation in the expression. The operation here is division: \(9 \div 3\). Implementing this operation will give the final value of the expression.

Key Concepts

Substituting Variables

When faced with an algebraic expression, it is often necessary to replace the variables with their given values, a process known as substituting variables. Imagine you have a recipe that calls for a certain number of ingredients; substituting variables is like replacing 'a cup of sugar' with the actual sugar you have on your kitchen counter.

For example, in the expression \(\frac{9}{p}\), if we are told that \(p = 3\), we would substitute \(3\) for every \(p\) in the expression. Thus, \(\frac{9}{p}\) becomes \(\frac{9}{3}\). Substitution is the first step in evaluating an expression, offering a clear path forward to find the expression's value in a concrete sense.

Simplifying Expressions

Simplifying an expression means to make it as basic as possible. It's like cleaning up your room so that it’s easier to navigate. After substituting \(3\) for \(p\), we get \(\frac{9}{3}\). The next step is to 'clean up' the expression or simplify it.

This process often involves arithmetic operations like addition, subtraction, multiplication, and division. By simplifying \(\frac{9}{3}\), we divide 9 by 3, which simplifies to 3. Simplification helps in understanding the core value of an expression without the clutter of unnecessary mathematical symbols.

Algebraic Operations

Algebraic operations are the building blocks of algebra and include addition, subtraction, multiplication, and division. When working with algebraic expressions, applying these operations correctly allows us to manipulate and solve equations effectively.

In our substitution example, we employed division, one of the fundamental algebraic operations. After substituting the variables, these operations are used to reach a simpler form or the final answer of the expression, showcasing the power of algebra to transform complex expressions into understandable results.

Division in Algebra

Division is a core component of algebraic operations, and it's particularly important when simplifying expressions. It is akin to breaking down a big box into smaller, more manageable pieces. In algebra, division helps us to find out how many times a number (divisor) is contained within another number (dividend).

In the given problem, division is used to evaluate the expression \(\frac{9}{3}\) after substituting \(p\) with \(3\). The division of 9 by 3 results in 3, thereby simplifying the expression down to a single number. Understanding division within the context of algebra is crucial because it helps to reveal the relationship between numbers, and it often serves as a step toward solving more complex equations.