Problem 26 - College Algebra

Question

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

Step-by-Step Solution

Answer

The evaluated expression for \( |x-y| \) with given values \( x=2 \) and \( y=-5 \) is 7.

Step 1: Substitute the values of x and y into the expression

The first step is to substitute the provided values for x and y into the expression \(|x-y|\). So the expression becomes \(|2 - (-5)|\).

Step 2: Simplify the expression inside the absolute value

The next step involves simplifying the expression inside the absolute value sign. The expression inside the absolute value sign is \(2-(-5)\), which simplifies to \(2+5 = 7\). So our expression now becomes \(|7|\).

Step 3: Evaluate the absolute value

Now, the absolute value of 7 needs to be evaluated. The absolute value of a number is the non-negative value of that number, regardless of its sign. So, the absolute value of 7 is just 7.

Key Concepts

Absolute Value

The concept of absolute value is quite straightforward once you get the hang of it. Simply put, the absolute value of any number is its distance from zero on the number line. It is always a non-negative number, which means you drop any negative sign that might be in front of the number.
For instance:

• The absolute value of 5 is 5, written as \(|5| = 5\).
• Similarly, the absolute value of -5 is also 5, written as \(|-5| = 5\).

This idea is consistent across all numbers, so whenever you're working with absolute values, remember that you're focusing only on the size of the number, not its sign. For example, when we evaluated the expression \(|7|\), we saw that the absolute value remained 7. This is because 7 is already a positive number.

Substitution Method

Substitution is a vital method in algebra. It allows us to solve expressions or equations by 'plugging in' values for the variables given in any problem. This method is especially helpful because it simplifies the process of working with variable-heavy equations.
In the original exercise, you were asked to evaluate \(|x-y|\) when \(x=2\) and \(y=-5\). By substituting, we replace each instance of 'x' with 2 and 'y' with -5, making complex algebraic expressions far more manageable.
For example:

• Start with the expression \(|x-y|\).
• Substitute the variables: \(|2-(-5)|\).
• Then, simplify further to get a clearer numerical expression.

Using substitution effectively reduces the difficulty of problems and can guide you to an accurate solution.

Simplification of Expressions

Simplification in algebra involves breaking down complex expressions into simpler or more manageable forms. This is a necessary skill as it is used in almost every mathematical problem-solving process.
In our solution, simplification involved a critical step. After substituting for variables, we dealt with the expression \(2-(-5)\).
Here’s how simplification works:

• Recognize that subtracting a negative is equivalent to addition. Thus \(2-(-5)\) becomes \(2+5\).
• Compute this addition: \(2+5=7\).
• Finally, our simplified expression under the absolute value was \(|7|\).

Simplifying ensures clarity and often allows you to see the solution path better. Remember, algebra is all about making intricate expressions easier to handle.