Problem 58 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 14.

Step 1: Identify the Expression

The expression given is \( yz - x \). We need to evaluate this expression by substituting the values of \(x\), \(y\), and \(z\) into it.

Step 2: Substitute the Known Values

Substitute \(x = 1\), \(y = 3\), and \(z = 5\) into the expression. This changes the expression from \( yz - x \) to \( 3 \times 5 - 1 \).

Step 3: Calculate the Multiplication

Calculate the multiplication part of the expression first: \( 3 \times 5 = 15 \).

Step 4: Perform the Subtraction

Subtract \(1\) from the result of the multiplication: \(15 - 1 = 14\).

Step 5: Compile the Result

The calculated result of the expression \( yz - x \) when \(x=1\), \(y=3\), and \(z=5\) is \( 14 \).

Key Concepts

Substitution Method

When dealing with algebraic expressions, the substitution method is a powerful tool for evaluating them. It involves replacing variables in an expression with their given numerical values. This is particularly useful when an expression involves multiple variables and you're tasked with finding an exact numerical result.

The process is straightforward:

• Identify the variables within your expression that need values substituted.
• Determine the numerical values that each variable represents.
• Replace each variable in the expression with its corresponding value.

This method simplifies complex expressions into basic arithmetic operations, allowing for easy calculation. For example, in the exercise, we substituted \(y = 3\) and \(z = 5\) into the expression \(yz\), transforming it into \(3 \times 5\), and substituting \(x = 1\) afterwards made the final arithmetic manageable.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators such as addition, subtraction, multiplication, and division. Unlike an equation, it doesn't assert equality but represents a value based on the variables it contains.

In the given exercise, the algebraic expression is \(yz - x\). Each part of the expression has a specific role:

• \(yz\) represents the multiplication of the variables \(y\) and \(z\).
• \(- x\) indicates that the result of \(yz\) will be reduced by the value of \(x\).

Understanding the structure of algebraic expressions helps in recognizing which operations to perform when evaluating them. It's crucial to perform these operations in the correct order, as determined by the fundamental arithmetic rules.

Arithmetic Operations

In mathematics, arithmetic operations create the backbone of computations. They include addition, subtraction, multiplication, and division, and each has its own set of rules, particularly concerning the order of operations. When evaluating expressions, mastering these rules is essential.

The expression \(yz - x\) primarily uses multiplication and subtraction. According to the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), multiplication is performed before subtraction:

• First, multiply \(y\) and \(z\): \(3 \times 5 = 15\).
• Next, subtract \(x\) from the result: \(15 - 1 = 14\).

These operations follow a linear sequence dictated by PEMDAS, ensuring the expression is simplified accurately. Taking care with the order of operations prevents errors and ensures the final result is correct, as demonstrated in the exercise.