Problem 37 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-2(-6 x+y-2 z), \text { if } x=1, y=1, \text { and } z=2$$

Step-by-Step Solution

Answer

The value of the expression is 18.

Step 1: Substitute the Given Values

First, substitute the given values of \(x = 1\), \(y = 1\), and \(z = 2\) into the expression \(-2(-6x + y - 2z)\). This gives:\[-2(-6(1) + 1 - 2(2))\]

Step 2: Simplify Inside the Parentheses

Evaluate the expression inside the parentheses:\(-6(1) + 1 - 2(2)\)= \(-6 + 1 - 4\).

Step 3: Calculate the Result of the Parentheses

Carry out the arithmetic:\(-6 + 1 - 4 = -9\).

Step 4: Apply the External Multiplication

Multiply the result from inside the parentheses by \(-2\), using the expression:\(-2 \times (-9)\).This gives you \(18\).

Key Concepts

Understanding Expressions

Expressions in algebra are combinations of variables, numbers, and operations. They represent a single value when given specific variables. In our example, the expression is \[ -2(-6x + y - 2z) \]. Key parts include:

• Variables: Like \(x\), \(y\), and \(z\), which are placeholders that can represent different values.
• Constants: These are the numbers themselves like \(-6\) and \(-2\).
• Operations: Mathematical functions such as addition (+) and multiplication (×) that combine the numbers and variables together.

Expressions need to be simplified correctly by following the proper order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This structure ensures each part of the expression is evaluated in the right sequence.

Mastering Arithmetic Operations

Arithmetic operations are basic math processes that help us solve expressions. They include addition, subtraction, multiplication, and division. In our expression, we dealt with a combination of these:

• Substitution: Replacing variables with their given values.
• Multiplication: For example, multiplying \(-6\) by \(1\) and multiplying \(-2\) by \(2\).
• Addition/Subtraction: Adding 1 to \(-6\) and then subtracting 4 results in more compact calculations, like \((-6 + 1 - 4)\).

Performing these operations correctly and in sequence is crucial. Only after simplifying the expression within the parentheses should external operations (like multiplication outside parentheses) be applied. This flow of steps helps maintain accuracy in solving algebraic expressions.

Handling Negative Numbers

Negative numbers are values below zero, and they play a significant role in arithmetic operations. They must be handled carefully to avoid errors, especially when combined with other operations. In our example, here's how negative numbers were managed:

• Multiplying two negative numbers results in a positive product, such as when multiplying \(-2\) by \(-9\), giving \(18\).
• Adding and subtracting negatives can sometimes confuse, but visualizing them on a number line can help. Moving left represents negatives, and moving right represents positives.
• Always keep track of the signs in operations. Mistakes in signs can lead to incorrect results.

By understanding how to operate negative numbers properly, solving complex expressions becomes a smoother process.