Problem 53 - Algebra 2

Question

Evaluate each expression if \(x=-2, y=6,\) and \(z=5\) $$ 2 x-y+3 z $$

Step-by-Step Solution

Answer

The value of the expression is 5.

Step 1: Substitute the Given Values

First, replace the variables in the expression with the given values: replace \(x\) with \(-2\), \(y\) with \(6\), and \(z\) with \(5\). The expression becomes: \[2(-2) - 6 + 3(5)\]

Step 2: Perform Multiplication

Next, perform the multiplication within the expression. Multiply \(2\) by \(-2\) to get \(-4\) and \(3\) by \(5\) to get \(15\):\[-4 - 6 + 15\]

Step 3: Execute Addition and Subtraction

Now, perform the addition and subtraction from left to right. Start by subtracting \(6\) from \(-4\) which gives \(-10\), then add \(15\) to \(-10\) to get \(5\):\[-10 + 15 = 5\]

Key Concepts

The Substitution Method

The substitution method is a handy tool for evaluating expressions. It involves replacing variables with their actual values so you can perform calculations directly. This process simplifies expressions and makes it easier to find a numerical answer. Here's how it works:

• Identify the variables in your expression and the values they are supposed to represent. In our example, we have variables \(x = -2\), \(y = 6\), and \(z = 5\).
• Replace each variable in the expression with its corresponding value. This turns the expression from something like \(2x-y+3z\) into \(2(-2)-6+3(5)\).

By using the substitution method, you are essentially simplifying your expression into terms you can easily calculate. This step sets the stage for using the order of operations to find the final result.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed. This is crucial for ensuring that everyone calculates expressions the same way and arrives at the same result.

Here's a quick rundown of the order of operations:

• Parentheses: Complete any calculations inside parentheses first.
• Exponents: Evaluate exponents or powers next.
• MD - Multiplication and Division: Perform these operations from left to right.
• AS - Addition and Subtraction: Finally, execute these from left to right.

"Please Excuse My Dear Aunt Sally" is a mnemonic to remember this sequence (Parentheses, Exponents, MD - Multiplication and Division, AS - Addition and Subtraction).

In the example expression \(2(-2)-6+3(5)\), we first perform the multiplication: \(2 \times -2 = -4\) and \(3 \times 5 = 15\). Only then do we move on to addition and subtraction, leading us to the final result.

Integer Operations

Integer operations refer to calculations involving whole numbers, which include positive numbers, negative numbers, and zero. Understanding how to work with integers is crucial because it lays the foundation for solving many mathematical problems.

When dealing with integer operations, you might encounter:

• Addition and Subtraction: When adding a positive number, move to the right on a number line. When you subtract, you move left.
• Multiplication and Division: Multiplying two positive numbers or two negative numbers gives a positive result, while multiplying a positive number by a negative one gives a negative result.

In our evaluated expression \{-4 - 6 + 15\}, we see a mix of integer operations. Here's a brief idea of their application:

- Start with \(-4 - 6\) which becomes \(-10\) (moving further left on the number line).
- Then, \(-10+15\) shifts 15 units right, landing you at \(5\).

Getting comfortable with integer operations will make manipulation of numbers in algebra and beyond much easier!