Problem 63 - College Algebra Essentials

Question

Evaluate each algebraic expression for x = 2 and y = -5. $$|x|+|y|$$

Step-by-Step Solution

Answer

The evaluated expression equals 7.

Step 1: Substitute the Given Values

First, substitute the given values into the equation: Replace x with 2, and y with -5: \[|2|+|-5|\]

Step 2: Evaluate the Absolute Values

Next step is to evaluate the absolute values: Absolute value of 2 is 2, absolute value of -5 is 5. Thus, the equation simplifies to: \[2+5\]

Step 3: Find the Sum

Finally, find the sum: \[2+5=7\]

Key Concepts

Absolute Value

In mathematics, the absolute value of a number represents its distance from zero on a number line, regardless of direction. It is denoted by two vertical lines encompassing the number, such as \( |x| \). The absolute value of a positive number or zero is the number itself, and the absolute value of a negative number is its positive counterpart.

For instance, the absolute value of 2 is 2 (\( |2| = 2 \) ), and the absolute value of -5 is 5 (\( |-5| = 5 \) ). When dealing with absolute values in algebra, it's important to treat them as a shielded quantity during operations like addition or multiplication until you've evaluated what's inside the shield – that is, until you've found the non-negative value they represent.

Substitution Method

The substitution method is a fundamental approach in algebra used to simplify expressions or solve equations. It involves replacing variables with numbers or other expressions to evaluate or further simplify the expression. The main goal is to break down complex problems into more manageable pieces.

In our exercise, we used the substitution method to replace 'x' with 2 and 'y' with -5 (\( x = 2, y = -5 \)), turning the algebraic expression into one with actual numbers that we can calculate with. It is essential to perform substitution carefully to avoid mistakes such as changing signs or improperly applying operations to the substituted values.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions, featuring one or more variables. These equations form the basis of algebra and allow us to describe relationships, solve for unknowns, and model real-world scenarios.

The expression \( |x| + |y| \) isn't an equation since it doesn't have an equals sign; instead, it's referred to as an 'algebraic expression'. However, understanding how to manipulate algebraic expressions using rules that apply to equations is crucial. When working with these expressions, such as in our exercise example, the goal is to evaluate them to find their simplest form or, if applicable, solve them when they are set equal to another expression or number.