Problem 55 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

11

Step 1: Substitute the given values

First, substitute the given values into the expression. We have \(x = 1\), \(y = 3\), and \(z = 5\), but \(z\) is irrelevant to this particular expression. The expression is \[ |2x + 3y|. \]Substituting the values gives us\[ |2(1) + 3(3)|. \]

Step 2: Simplify inside the absolute value

Now, simplify the expression inside the absolute value:\[ 2(1) + 3(3) = 2 + 9 = 11. \]

Step 3: Evaluate the absolute value

The expression now becomes \[ |11|. \]Since 11 is positive, the absolute value is simply \[ 11. \]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition or multiplication). They represent the relationship between different quantities. In the exercise, the expression \(|2x + 3y|\) involves variables \(x\) and \(y\), numbers 2 and 3, and mathematical operations of multiplication and addition.

The goal when manipulating algebraic expressions can vary from simple evaluation to solving equations. Here we focus on evaluating the expression given specific values for the variables.

Key components of algebraic expressions include:

• **Variables:** Symbols (like \(x\) and \(y\)) representing numbers.
• **Coefficients:** Numbers used to multiply the variables (like 2 and 3 in our example).
• **Constants:** Numbers on their own without any associated variable.
• **Operators:** Signs that indicate the operations to perform (like the "+" sign).

Understanding the roles of these elements is crucial for manipulating expressions correctly.

Substitution Method

The substitution method involves replacing the variables in an algebraic expression with specific values. This is an essential technique for evaluating expressions, as seen in the given exercise.

Here's the step-by-step approach for substitution:

• **Identify all variables:** Pinpoint what each variable represents. For example, in \(|2x + 3y|\), we identify the variables \(x\) and \(y\).
• **Replace variables with values:** Substitute the variables with their given values. For \(x = 1\) and \(y = 3\), we replace \(x\) and \(y\) in the expression, turning it into \(|2(1) + 3(3)|\).
• **Follow mathematical operations:** Perform the necessary operations within the expression after substitution.

This method ensures that what's being evaluated reflects the values represented by each variable, allowing for straightforward computation.

Evaluating Expressions

Evaluating expressions involves finding the value of an algebraic expression by carrying out the operations after substituting variables with given numbers. This process is fundamental in both algebra and calculus, as it turns an abstract expression into a concrete number.

For the exercise at hand, once the given values were substituted into \(|2x + 3y|\), the expression becomes \(|2(1) + 3(3)|\). By simplifying this, we perform the arithmetic inside the absolute value:

• **Multiplication:** Compute each product, \(2 \times 1 = 2\) and \(3 \times 3 = 9\).
• **Addition:** Sum the results, \(2 + 9 = 11\).

Finally, evaluate the absolute value to find the result. Since 11 is already positive, \(|11| = 11\). Hence, the evaluated expression results in 11.

Evaluating expressions is pivotal in converting mathematical ideas into tangible results.