Problem 13 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$z+2$$

Step-by-Step Solution

Answer

The value of the expression is 11.

Step 1: Identify the Variables

In this expression, we are given that variables have specific values: \( z = 9 \), \( x = 7 \), and \( y = 3 \). The expression we need to evaluate is \( z + 2 \). Since the expression involves \( z \), we will use its value.

Step 2: Substitute the Variable

Replace the variable \( z \) in the expression \( z + 2 \) with its given value. So, the expression becomes \( 9 + 2 \).

Step 3: Perform the Calculation

Add the numbers after substitution. Calculate \( 9 + 2 = 11 \).

Key Concepts

Substitution Method

In algebra, the substitution method is a powerful tool for simplifying expressions and solving equations. Imagine it as swapping numbers in place of variables to make calculations straightforward. The process begins with identifying which variables in the given expression have known values. For the exercise here, you're introduced to three variables, namely \(x\), \(y\), and \(z\), each with given values.

• \(x = 7\)
• \(y = 3\)
• \(z = 9\)

The task is to substitute these values into the expression where the variables appear. In our example expression, \(z + 2\), we only need the value of \(z\), which is 9. By replacing \(z\) with 9, the expression transforms to a simple arithmetic calculation: \(9 + 2\). This substitution makes the expression easy to handle and leads to an immediate solution.

Basic Arithmetic Operations

Once the substitution has been made, the next step involves basic arithmetic operations. These operations are the foundation of mathematics and include addition, subtraction, multiplication, and division. In our exercise, after substituting \(z = 9\), we are left with the expression \(9 + 2\). This reduction means we need to perform an addition operation.

• Addition: Combining numbers to get a sum.

For addition, you simply take the numbers provided—here, 9 and 2—and combine them to find the total. If you add 9 to 2, you get 11. Arithmetic operations are crucial because they help you solve and simplify expressions and are used in almost all math-related tasks. They are like the main ingredients in a recipe for solving mathematical problems.

Variables in Algebra

Variables are essentials in algebra, serving as placeholders for unknown or varying values that can change depending on the problem. In any algebraic expression, variables are represented by letters, such as \(x\), \(y\), and \(z\) in this exercise.

• \(x\) could stand for another quantity in different contexts.
• The value of \(y\) could vary in a different problem.
• \(z\) might be something else in another exercise.

Understanding variables involves recognizing their role in expressions and equations. They help generalize mathematical concepts, allowing unique solutions to specific cases when unique values are substituted, as seen in this example. Without variables, algebra as a structured mathematical branch would not exist as we know it. Recognizing how to manipulate and substitute them is essential for progressing in math.