Problem 51 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(3 x-2\)

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Substitute the values

Replace the variable in the expression with its corresponding value. In this expression, substitute \(x\) with 1. Thus, the expression \(3x - 2\) becomes \(3(1) - 2\).

Step 2: Calculate the multiplication

Perform the multiplication operation. In this case, multiply 3 and 1. This gives us \(3 \times 1 = 3\).

Step 3: Subtract to find the result

Subtract 2 from the result of the multiplication. This is calculated as \(3 - 2 = 1\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra used to replace variables with given numerical values. This approach simplifies expressions and aids in solving equations. For instance, in the algebraic expression presented, we replace the variable \(x\) with a number provided, which is 1. This is like filling in the blank with the correct number.

• Identify the expressions that contain variables.
• Replace each variable with its corresponding value.
• This method can be helpful in simplifying expressions or solving equations.

Using substitution helps make abstract numerical relationships concrete by converting expressions into arithmetic calculations.

Algebraic Expressions

Algebraic expressions are mathematical statements that include variables, numbers, and operation symbols (like addition, subtraction, multiplication, etc.). These expressions represent relationships between numbers.
In our case, the expression is \(3x - 2\). The variable \(x\) is multiplied by 3 and then reduced by 2.

• Variables: Stand-in symbols that can represent unknown or variable quantities.
• Coefficients: Numbers that multiply variables (like 3 in \(3x\)).
• Constants: Numbers that stand alone (like the -2).

Understanding how to manipulate these expressions is crucial for solving algebraic problems efficiently.

Order of Operations

The order of operations is a set of rules outlining the sequence to follow when evaluating mathematical expressions. Remembering this sequence helps avoid mistakes when dealing with complex expressions.A common acronym, PEMDAS, stands for:

• Parentheses: Complete calculations inside parentheses first.
• Exponents: Evaluate exponents next.
• Multiplication and Division: Process these from left to right.
• Addition and Subtraction: Lastly, perform these operations from left to right.

In the expression \(3x - 2\), once the substitution is done, you first handle the multiplication (3 times 1) and then perform the subtraction (minus 2) following the order prescribed by PEMDAS. This prioritization ensures accuracy in simplifying and solving problems.