Problem 36 - Algebra 2

Question

Write an algebraic expression in which subtraction is performed before division, and the symbols ( ), [ ], or { } are not used.

Step-by-Step Solution

Answer

Expression: \( x - y \,/ \, z \) ensures subtraction before division.

Step 1: Understand the Problem

To solve this problem, we need to create an expression where subtraction is performed before division naturally, without using any grouping symbols like parentheses, brackets, or braces.

Step 2: Use the Order of Operations

Recall that, according to the order of operations (PEMDAS/BODMAS), division should usually come before subtraction unless grouping symbols dictate otherwise. Since we cannot use any grouping symbols, we have to find a way to create an expression where subtraction naturally occurs first.

Step 3: Write the Expression

Consider the expression \( x - y \,/ \, z \). According to PEMDAS/BODMAS, this expression would naturally evaluate the subtraction first. This happens because this expression is equivalent to \( \frac{x - y}{z} \), which implicitly involves computing \( x - y \) first as it forms the numerator, thus ensuring subtraction is done prior to division.

Key Concepts

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the sequence in which operations should be performed to accurately solve an expression. Without a clear set of rules, expressions could be misinterpreted, leading to incorrect results. In general, the order of operations ensures consistency in mathematical communication and calculation.
*Operations and their hierarchy:*

• Start with operations inside grouping symbols: parentheses (), brackets [] and braces {}
• Exponents (such as powers and square roots)
• Multiplication and Division, performed from left to right
• Addition and Subtraction, also from left to right

In this exercise, understanding and manipulating the order of operations is key, as we aim to perform subtraction before division naturally, even without grouping symbols like parentheses.

PEMDAS/BODMAS

PEMDAS and BODMAS are acronyms that represent the sequence of operations we follow in mathematical expressions. The acronyms stand for:
*PEMDAS:*

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

*BODMAS:*

• B: Brackets
• O: Orders (exponents and roots)
• D: Division
• M: Multiplication
• A: Addition
• S: Subtraction

By following this order, we can ensure that our calculations are accurate. In PEMDAS/BODMAS, multiplication and division are of equal priority; they are done from left to right. The same applies to addition and subtraction. Our challenge was to perform subtraction before division without grouping symbols, and by the nature of the expression \( x - y / z \), the subtraction operation in the numerator \( x - y \) takes precedence.

Mathematical Notation

Mathematical notation is the language through which we express and communicate complex mathematical ideas clearly and succinctly. Notations include numbers, symbols, and operators, arranged in a conventional form that mathematicians and learners alike can understand.
*A few key elements of mathematical notation include:*

• Numbers and variables, like \( x \), \( y \), and \( z \)
• Operators, such as addition \( + \), subtraction \( - \), multiplication \( \times \), and division \( / \)
• Fractional notation, such as \( \frac{a}{b} \), which implies dividing \( a \) by \( b \)

In our situation, the expression \( x - y / z \) cleverly uses mathematical notation to enforce subtraction before division. Here, the subtraction is interpreted as taking precedence since it involves creating a fraction, showcasing the power of notation to influence the order of operations without explicit grouping symbols.