91Ó°ÊÓ

Understanding the concept of the absolute value is essential when working with numbers on a number line. Absolute value refers to the distance of a number from zero, disregarding which direction—left for negatives or right for positives—the number is on this line. It's represented by two vertical lines, like this: \( |a| \). For any real number \( a \) , its absolute value is presented as \( |a| \) and is always non-negative.

For example, the absolute value of both \( -3 \) and \( 3 \) is \( 3 \) because both are three units away from zero. Hence, when you see an expression like \( |x-y| \) in mathematics, it is asking for the distance between \( x \) and \( y \) regardless of the order of subtraction. This becomes particularly useful when dealing with real world quantities that cannot be negative, such as distances or magnitudes.

The substitution method is a fundamental part of algebra which involves replacing variables with numbers or other expressions. This is a direct approach to solving algebraic expressions and equations. When you're given values for variables, like \( x=2 \) and \( y=-5 \), you substitute these values into the expression wherever those variables appear.

To properly use the substitution method, follow these steps:

• Identify the variables in the expression.
• Determine which values correspond to the variables.
• Replace (substitute) the variables with the given numbers.
• Carefully perform the algebraic operations in the correct order after the substitution.

Substitution is pivotal not only for evaluating expressions but also when solving systems of equations, where you solve for one variable in terms of the others and then substitute this value into another equation.

Algebraic operations include addition, subtraction, multiplication, division, and sometimes more advanced operations such as exponentiation, root extraction, or working with logarithms. When working with algebraic expressions, it's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

For the given exercise \( |x-y| \) with \( x=2 \) and \( y=-5 \) , the operation starts with subtraction \( 2 - (-5) \) followed by evaluation of the absolute value. The subtraction inside the absolute value results in addition (a double negative turns into a positive), leading to \( 2 + 5 \) which equals \( 7 \). The absolute value of \( 7 \) is \( 7 \) as it is already a positive number. Remember, multiplication and division are performed before addition and subtraction unless grouped by parentheses.