Problem 30 - Fundamentals of Mathematics

Question

Find the value of each expression. $$2 x-3 y+4 z, \text { if } x=1, y=-1, \text { and } z=-2$$

Step-by-Step Solution

Answer

The value of the expression is -3.

Step 1: Identify Given Values

The problem provides specific values for the variables in the expression. We need to substitute these values into the expression: \(x = 1\), \(y = -1\), and \(z = -2\).

Step 2: Substitute Values into Expression

Replace each variable in the expression \(2x - 3y + 4z\) with the given values: \(x = 1\), \(y = -1\), and \(z = -2\). The expression becomes \(2(1) - 3(-1) + 4(-2)\).

Step 3: Perform Multiplication

Calculate each term separately by multiplying the constants with their respective variables: 1. \(2 \times 1 = 2\)2. \(-3 \times (-1) = 3\)3. \(4 \times (-2) = -8\).

Step 4: Simplify the Expression

Combine the calculated terms: \(2 + 3 - 8\).

Step 5: Calculate the Sum

Add the first two terms and then subtract the last term: \(2 + 3 = 5\)Then, \(5 - 8 = -3\).

Key Concepts

Expression Evaluation

In algebra, evaluating an expression involves substituting given numeric values into the variables and simplifying to find a single numerical result. This process is fundamental as it translates the abstract symbols into concrete numbers, making the algebraic concept practical. When we substitute values into expressions, we must follow the correct arithmetic operations and order. For instance, in the expression \(2x - 3y + 4z\), the values for \(x\), \(y\), and \(z\) are provided as \(1\), \(-1\), and \(-2\) respectively. Substituting these values directly into the expression changes it into a simple form: \(2(1) - 3(-1) + 4(-2)\). After substitution comes evaluation. Evaluate step by step: first perform multiplications, then additions or subtractions as per arithmetic rules. These steps ensure that all calculations follow the mathematical order of operations (PEMDAS/BODMAS). This systematic approach not only makes evaluating expressions easier but also prevents errors. Thus, evaluating expressions is a vital skill that involves understanding how to plug values into variables and simplify the operations properly. This creates clarity and helps us solve algebraic problems effectively.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and sometimes division. These expressions form the building blocks of algebra and can represent various mathematical relationships. In the expression \(2x - 3y + 4z\), the components are:

• **Numbers (constants):** These are the coefficients like 2, -3, and 4, which multiply the variables.
• **Variables:** Letters like \(x\), \(y\), and \(z\) that stand for unknown values or can be assigned particular numbers.
• **Operators:** These are the plus (+) and minus (-) signs indicating the operations to be performed.

Understanding the structure of algebraic expressions helps in manipulating them for various purposes such as solving equations or evaluating expressions. These expressions are crucial as they allow abstract ideas to be conveyed in a concise form. They model real-world phenomena where exact numbers may not be known but relationships and potential changes can be defined algebraically.

Mathematical Operations

Mathematical operations form the core of algebra and involve basic procedures like addition, subtraction, multiplication, and occasionally division. Knowing the correct order to perform these operations ensures accuracy when working with expressions. When evaluating the expression \(2x - 3y + 4z\), it’s crucial to execute operations in the correct sequence:- **Multiplication:** First, multiply each variable by its corresponding coefficient as it is a priority operation. - For instance: \(2 \times 1 = 2\) \(-3 \times (-1) = 3\) \(4 \times (-2) = -8\)- **Addition and Subtraction:** Next, perform any addition or subtraction in the order they appear from left to right. - Combine: \(2 + 3 - 8\). - Adding gives \(5\) and subtracting results in \(-3\).This proper order is key in simplifying expressions accurately and effectively. Remember, **PEMDAS/BODMAS** is the general rule—Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following these guidelines assures the correctness of the calculations and results.