Problem 16 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$11 w+3 x-6 w-5 w+8 x-11 x$$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Identify Like Terms

First, we need to identify terms that can be combined. In this expression, terms with the same variable can be combined. We have terms with 'w': \(11w, -6w, -5w\) and terms with 'x': \(3x, 8x, -11x\).

Step 2: Combine Like Terms for 'w'

Let's combine the terms with 'w'. We have: \(11w - 6w - 5w\).Combine these terms: \[11w - 6w - 5w = (11 - 6 - 5)w = 0w.\]So, the terms with 'w' cancel out and result in 0.

Step 3: Combine Like Terms for 'x'

Now, let's handle terms with 'x'. We have: \(3x + 8x - 11x\).Combine these terms:\[3x + 8x - 11x = (3 + 8 - 11)x = 0x.\]So, the terms with 'x' also cancel out and result in 0.

Step 4: Simplified Expression

Since all terms have canceled each other out, the final simplified expression is 0.

Key Concepts

Variable Identification

Identifying variables in an algebraic expression is a crucial first step to any kind of simplification or solving process. Variables are essentially symbols, often letters like \(w\) or \(x\), that represent unknown numbers. Spotting them in expressions helps in organizing and simplifying our work. In our example expression, we have variables \(w\) and \(x\). They appear with coefficients, which are numbers that multiply the variables. Like terms are terms in an expression that have the same variable raised to the same power. This makes it possible to add or subtract them directly.
Identifying variables and recognizing like terms quickly becomes a key skill as you progress in mathematics:

• Look for repeating variable symbols like \(w\), \(x\), etc.
• Check the powers of these variables - they must be the same for terms to be "like".
• Note down their coefficients for easy combination in the next steps.

This foundational skill will aid in simplifying expressions efficiently.

Expression Simplification

Simplifying expressions is the process of making an algebraic expression as compact as possible while retaining its value. The focus is on combining like terms to reduce the clutter. In our given problem, by collecting like terms and combining them, we reduce the expression to its simplest form.
Let's break down the process further:

• Collect like terms together, in our case, terms involving \(w\) and terms involving \(x\).
• Combine the terms by adding or subtracting the coefficients. For instance, combining the \(w\) terms: \(11w - 6w - 5w = 0w\).
• Repeat the process for the next set of like terms, i.e., the \(x\) terms \((3x + 8x - 11x = 0x)\).

Simplifying does not change the value of the expression; rather, it presents the same value in a tidier format. The power of simplification shines through when dealing with more complicated equations, aiding in easier data interpretation and result realization.

Algebra

Algebra is a fundamental branch of mathematics, focusing on solving equations and understanding relationships between variables. It uses symbols and letters to represent numbers and operations in equations or expressions. Combining like terms is a strategy used frequently within algebra. One especially relevant task is simplifying expressions, as seen in our exercise.
Why is algebra important?

• It lays the groundwork for higher-level mathematics and sciences, where complex equations need to be simplified or solved.
• Understanding algebra enhances problem-solving and critical thinking abilities, as it involves identifying patterns and structures.
• Algebra is applied in various real-world contexts, from calculating interests to computer algorithms.

As you master algebraic techniques, like combining like terms, you'll find them valuable in countless scenarios where clear, logical, and structured thinking is required.