Problem 62 - Algebra and Trigonometry

Question

Evaluate each algebraic expression for \(x=2\) and \(y=-5\) $$|x-y|$$

Step-by-Step Solution

Answer

The absolute value of \(x - y\) when \(x = 2\) and \(y = -5\) is \(7\).

Step 1: Substitute the values

Substitute \(x=2\) and \(y=-5\) into \(|x-y|\) to get \(|2 - (-5)|\).

Step 2: Simplify the expression inside the absolute value

Simplify \(2 - (-5)\) to get \(2 + 5 = 7\)

Step 3: Apply the absolute value rules

Since the absolute value of a number is its distance from zero on the number line, regardless of direction, the absolute value of \(7\) will simply be \(7\). Thus \(|7| = 7\).

Key Concepts

Understanding Absolute Value

When we talk about the absolute value of a number, we're discussing its distance from zero on a number line, regardless of the direction. It's denoted by two vertical bars surrounding the number. For instance, the absolute value of both -3 and 3 is 3, because each is three units away from zero. In algebraic expressions, finding the absolute value is a two-step process. First, you evaluate the expression inside the vertical bars. Then, irrespective of the sign of the resulting number, you take its positive value. In our example, evaluating the expression for the given values of x and y leads to |2 - (-5)|, which simplifies to |7|. Since 7 is already positive, its absolute value remains 7.

This concept is particularly important as it helps students understand numerical values in terms of their magnitude rather than just their algebraic sign. It is also the foundational concept when solving equations and inequalities involving absolute value, making it a staple in understanding more complex mathematical concepts.

The Substitution Method in Algebra

In the realm of algebra, the substitution method is a fundamental technique used to simplify expressions or solve equations. It involves replacing variables with their corresponding numerical values. To apply this method effectively, you first need to know the value of the variable you are replacing. In our exercise example, we are given that x=2 and y=-5. When we substitute these values into the algebraic expression |x-y|, we get |2 - (-5)|. This substitution makes the expression much easier to evaluate.

The substitution method is essential for solving problems across different areas of mathematics, including algebra, calculus, and beyond. It allows students to work with concrete numbers instead of variables, making abstract concepts more tangible and consequently easier to understand.

Simplifying Expressions

The process of simplifying expressions is a pillar in algebra. It involves reducing an expression to its simplest form, making it easier to understand or solve. Simplifying can involve combining like terms, applying the distributive property, or carrying out arithmetic operations. For our exercise, the expression initially contains a subtraction of a negative number—which is equivalent to addition. So, simplifying the expression inside the absolute value brackets, we convert 2 - (-5) to 2 + 5, giving us 7. The aim is to break down the expression into its most basic form without changing its value.

Mastering this process is crucial since it not only eases the path to solution for an immediate problem but also helps in understanding higher-level math where the simplification of complex expressions is an everyday task. By internalizing the steps and practicing regularly, students can quickly learn to recognize patterns and simplify expressions more intuitively.