Problem 24 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$16 x-12 y+5 x+7-5 x-16-3 y$$

Step-by-Step Solution

Answer

The simplified expression is \(16x - 15y - 9\).

Step 1: Identify Like Terms

Look at each term and identify which terms can be combined. In the expression \(16x - 12y + 5x + 7 - 5x - 16 - 3y\): \(16x\), \(5x\), and \(-5x\) are like terms. \(-12y\) and \(-3y\) are another set of like terms. Finally, \(7\) and \(-16\) are constants that can be combined.

Step 2: Combine the \(x\) Terms

Add together the coefficients of the like terms for \(x\): \(16x + 5x - 5x = (16 + 5 - 5)x = 16x\).

Step 3: Combine the \(y\) Terms

Add together the coefficients of the like terms for \(y\): \(-12y - 3y = (-12 - 3)y = -15y\).

Step 4: Combine the Constant Terms

Add together the constant numbers: \(7 - 16 = -9\).

Step 5: Write the Simplified Expression

Combine all the results from the previous steps to get the simplified expression: \(16x - 15y - 9\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a key technique to simplify expressions. Like terms are terms that have the same variables raised to the same power. The coefficients of these terms can differ, but the variables and their exponents must match exactly.
For instance:

• Like terms in the expression include:
  • \(16x\), \(5x\), and \(-5x\)
  • \(-12y\) and \(-3y\)
  • Constant terms like \(7\) and \(-16\)

To combine like terms, simply add or subtract the coefficients, and keep the variable and its power intact. This helps in organizing expressions, making them easier to work with and understand. It's like grouping similar items together in a collection. In our exercise, by combining the like terms step by step, we achieve a cleaner expression: \(16x - 15y - 9\).

Simplifying Expressions

Simplifying expressions in algebra involves reducing an expression to its most basic form. The goal is to make the expression as compact as possible while still maintaining its value and meaning. This often involves several processes such as:

• Combining like terms
• Performing arithmetic operations correctly

By combining like terms, as shown in the previous section, and carefully performing any arithmetic operations, we can transform a complex expression into a simplified form. This process is crucial for solving more complex problems later, as it provides a tidier version that's easier to manipulate or substitute into equations.
In our example, simplifying the expression \(16x - 12y + 5x + 7 - 5x - 16 - 3y\) led us to the cleaner version: \(16x - 15y - 9\). Now, it's ready for further calculations or integration into larger equations.

Coefficients in Algebra

A coefficient is a numerical or constant factor that multiplies a variable in an algebraic expression. Understanding coefficients helps in combining like terms and simplifying expressions since they dictate how terms can be combined. For example:

• In the term \(16x\), the number 16 is the coefficient.
• Coefficients dictate the weight or strength of a term with its variable.

When combining like terms, we focus on these numbers. It's important to add or subtract the coefficients properly to avoid errors. For example, while working through the term \(16x + 5x - 5x\), focus on the coefficients: \(16 + 5 - 5\) which gives you \(16x\).
Recognizing and handling coefficients correctly allows one to simplify complex expressions easily and lays the groundwork for advanced algebraic manipulations.