Problem 42 - Prealgebra

Question

Evaluate each expression if \(x=-12, y=4,\) and \(z=-1\) $$|y|-|x|+|z|$$

Step-by-Step Solution

Answer

The value of the expression is \\(-7\\).

Step 1: Understand Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of its sign. For example, the absolute value of \(4\) is \(|4| = 4\), and the absolute value of \(-12\) is \(|-12| = 12\). Similarly, \(|-1| = 1\).

Step 2: Substitute Known Values

Replace \(x\), \(y\), and \(z\) in the expression with their respective values: \(x = -12\), \(y = 4\), \(z = -1\). This gives us the expression \(|4| - |-12| + |-1|\).

Step 3: Calculate Absolute Values

Compute the absolute values for each part of the expression: 1. \(|4| = 4\). 2. \(|-12| = 12\). 3. \(|-1| = 1\).

Step 4: Perform Arithmetic Operations

Substitute the absolute values into the expression and perform the arithmetic operations: \(4 - 12 + 1\).1. Subtract: \(4 - 12 = -8\).2. Add: \(-8 + 1 = -7\).

Key Concepts

Substitution

Substitution is a useful technique in algebra. It involves replacing variables in an expression with their specified values. In the given exercise, we have three variables: \(x\), \(y\), and \(z\). Instead of keeping them as unknowns, we substitute \(x = -12\), \(y = 4\), and \(z = -1\).

By performing substitution, we update our initial expression from \(|y| - |x| + |z|\) to \(|4| - |-12| + |-1|\). This makes our task easier as we're working with concrete numbers instead of abstract variables. The clarity and simplicity substitution offers is key in simplifying and subsequently solving algebraic expressions.

Arithmetic Operations

Once substitution is complete, the next step is to carry out arithmetic operations on concrete numbers. This comprises addition, subtraction, multiplication, and division of numbers. In this particular problem, we have an arithmetic sequence of operations involving subtraction and addition.

Our main expression becomes: \(4 - 12 + 1\). We proceed with the operations as follows:

• First, perform the subtraction to find \(4 - 12\). This yields \(-8\).
• Next, perform the addition \(-8 + 1\), resulting in \(-7\).

It's essential to follow the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure each step is calculated correctly.

Evaluating Expressions

Evaluating expressions is a crucial aspect of algebra where you simplify them to find a solution. After substitution and executing arithmetic operations, you will get a final value, representing the solution of the expression.

For our case, after substituting \(x\), \(y\), and \(z\) with their given values and performing the necessary calculations, our expression evaluates to \(-7\).

Whenever you evaluate an expression, double-check each step, ensuring you transform and calculate correctly. Practicing this makes more complex expressions easier to handle as your skills and confidence grow. Understanding how to apply these foundational concepts ensures you navigate algebraic expressions effectively.