Problem 1 - Algebra 2

Question

Evaluate each expression if \(x=4, y=-2,\) and \(z=3.5\) \(z-x+y\)

Step-by-Step Solution

Answer

The evaluated expression is -2.5.

Step 1: Substitute Values into the Expression

From the provided expression \(z-x+y\), substitute the given values: \(z = 3.5\), \(x = 4\), and \(y = -2\). This gives us the expression \(3.5 - 4 + (-2)\).

Step 2: Perform First Operation: Subtract \(x\) from \(z\)

Calculate \(3.5 - 4\). This gives \(-0.5\) because 3.5 minus 4 results in a negative number.

Step 3: Add \(y\) to the Result

Now add \(-2\) to the previously obtained result of \(-0.5\). So the calculation is \(-0.5 + (-2) = -0.5 - 2 = -2.5\).

Key Concepts

Substitution

In mathematics, substitution is a fundamental concept used to replace variables with their actual values. This is often the first step in evaluating expressions or solving equations. In our example, we have the expression \(z-x+y\). To substitute, we simply replace each variable with its given value:

• \(z = 3.5\)
• \(x = 4\)
• \(y = -2\)

This results in a new expression: \(3.5 - 4 + (-2)\). Through substitution, we bridge the gap between abstract expressions and concrete numbers, making it easier to proceed with calculations.
Substitution ensures that every part of an expression is defined and ready for further mathematical operations. This step sets the stage for evaluating the expression effectively.

Order of Operations

The order of operations is a crucial rule in mathematics that dictates the sequence in which operations are performed. This rule is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression \(3.5 - 4 + (-2)\), we only have addition and subtraction, which are performed from left to right. This means we first evaluate \(3.5 - 4\), resulting in \(-0.5\), and then continue by addressing the addition of \(-2\) to this result.
Following the order of operations is essential for getting the correct solution. It's like following a recipe; skipping steps or rearranging them might result in a completely different outcome.

Arithmetic Operations

Arithmetic operations include the basic mathematical operations of addition, subtraction, multiplication, and division. These operations form the foundation of most mathematical calculations.
In our exercise, we primarily deal with subtraction and addition:

• First, subtract \(4\) from \(3.5\), which results in \(-0.5\). This negative result highlights the importance of understanding how subtraction can lead to negative numbers.
• Next, add \(-2\) to \(-0.5\), leading to \(-2.5\). Adding a negative number is essentially the same as subtracting a positive one, reinforcing the idea that addition and subtraction are interconnected.

Understanding these basic operations helps build confidence in managing and simplifying expressions. By mastering them, you can easily navigate through more complex mathematical problems.