Problem 99 - Algebra A Combined Function

Question

Perform each indicated operation. \(6 x+8 x\)

Step-by-Step Solution

Answer

The result of the operation is \(14x\).

Step 1: Identify Like Terms

In the expression given, \(6x + 8x\), both terms have the same variable \(x\), making them like terms. Like terms can be combined by adding their coefficients.

Step 2: Add the Coefficients

Add the numerical coefficients of the like terms. Coefficients are the numbers in front of the variable. In this case, let's add \(6\) and \(8\) to get \(14\).

Step 3: Combine Like Terms

Multiply the sum of the coefficients by the variable \(x\) to combine like terms: \(14x\).

Key Concepts

Coefficients

In a mathematical expression, coefficients are the numbers that are placed in front of variables. These numbers indicate how many units of the variable are involved in the expression. In the given problem, both terms, 6x and 8x, have coefficients. Here, "6" is the coefficient of the term 6x, and "8" is the coefficient of 8x. By understanding coefficients, you can determine the quantity of the variable being considered.

When you perform operations like addition or subtraction with expressions, coefficients make it easy to manage and simplify the process. They are simply numerical multipliers of variables, and when you have the same variables with different coefficients, you simply perform arithmetic operations on these coefficients. This is particularly helpful when combining like terms in expressions.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that simplifies expressions to make them easier to work with. Like terms are terms that have the same variable raised to the same power. For instance, in the expression 6x + 8x, both terms are like terms because they contain the variable x with no exponent or an exponent of 1. This means you can combine them.

Combining like terms involves adding or subtracting their coefficients and maintaining the same variable. So, when you see something like 6x + 8x, you add the coefficients 6 and 8, giving you 14. This results in a more simplified term, 14x. By combining like terms, you reduce the complexity of expressions, making them easier to understand and solve.

Variable

A variable is a symbol, typically a letter, that represents an unknown number or value in an expression or an equation. In the exercise provided, x is the variable. Variables are crucial in algebra because they allow us to write general formulas and solve problems even when the exact numbers aren't known.

In the expression 6x + 8x, the variable x represents the same unknown quantity in both terms. This is why these terms are considered like terms; they share the same variable. Variables give expressions flexibility and enable us to mold generic solutions that apply to a wide variety of specific cases. Understanding variables is essential for learning how to manipulate and solve algebraic expressions and equations.