Problem 55 - Prealgebra and Introductory Algebra

Question

Evaluate the variable expression for \(a=-2, b=3, c=-1,\) and \(d=4\) $$|a+d|^{2}+|c-b|^{2}$$

Step-by-Step Solution

Answer

The result of the expression for the given values is 20

Step 1: Substitute the Values

First, substitute the given values of the variables into the given expression: \( |(-2)+4|^2 + |-1-3|^2 \)

Step 2: Simplify inside the Absolute Values

Simplify the equations inside the absolute value symbols: \( |2|^2 + |-4|^2 \)

Step 3: Calculate the Absolute Values

An absolute value represents the distance of a number from zero, so it's always positive or zero: \( 2^2 + 4^2 \)

Step 4: Evaluate the Squares and Simplify

Simplify the expression by squaring the numbers and adding the results together: \( 4 + 16 \) which equals to 20

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on a number line, without considering direction. Essentially, it turns any negative number into a positive one. For example, the absolute value of \(-3\) is \(|-3| = 3\). This concept is vital when working with expressions containing absolute values because it ensures that the results are always non-negative.
In our exercise, after substituting the values, we simplify the absolute value expressions: \(|-2 + 4|\) and \(|-1 - 3|\). This results in \(|2|\) and \(|-4|\), which are 2 and 4, respectively.
Key aspects to remember about absolute values:

• Always returns a non-negative result.
• Represents a number's distance from zero on the number line.

Understanding these principles will help you apply the absolute value to any variable expression.

The Role of Substitution in Variable Expressions

Substitution is the process of replacing variables in an expression with given numerical values. This step is crucial because it allows us to transform an abstract expression into a calculable one. In the provided exercise, we substituted \(a = -2\), \(b = 3\), \(c = -1\), and \(d = 4\) into \(|a+d|^2 + |c-b|^2\).
Here's how substitution works:

• Start by identifying each variable in the expression.
• Replace each variable with its corresponding value.
• Simplify the expression with these numbers in place.

In our example, after substitution, the expression becomes \(|(-2) + 4|^2 + |-1 - 3|^2\). Understanding substitution helps in converting theoretical problems into real-world numbers, paving the way for simplification and evaluation.

Simplification: Breaking it Down

Simplification involves reducing an expression to its most basic form. After substitution, simplification assists in making the calculations more manageable and efficient. For our exercise, simplifying within the absolute values comes first: \(-2 + 4 = 2\) and \(-1 - 3 = -4\), which becomes \(|2|\) and \(|-4|\).
Simplify by considering:

• Perform operations inside absolute values as needed.
• Proceed step-by-step, handling one conversion or calculation at a time.

This methodical approach allows us to handle more complex expressions by always breaking them down into simpler parts. Simplification is your go-to tool for making variable expressions easier to work with.

Evaluating Expressions: Solving the Puzzle

Evaluating expressions is the process of solving them to find their numerical value. After substituting and simplifying the expression in our exercise, the next step is evaluation. Here, both absolute values were simplified to 2 and 4. Evaluate by squaring these values and adding the results: \((2)^2 = 4\) and \((4)^2 = 16\). Adding them together gives \(4 + 16 = 20\).
When evaluating expressions:

• Remember order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).
• Carry out operations systematically to avoid errors.

Mastering the evaluation of expressions ensures you can solve variable expressions accurately and efficiently. This skill enables you to tackle increasingly complex problems as you progress in mathematics.