Problem 17 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 6 x-5 x+x-3+2 x $$

Step-by-Step Solution

Answer

4x - 3

Step 1: Identify Like Terms

In the expression \(6x - 5x + x - 3 + 2x\), the like terms involving the variable \(x\) are \(6x\), \(-5x\), \(x\), and \(2x\). The constant term is \(-3\).

Step 2: Combine Like Terms with Variable

Add the coefficients of the \(x\) terms: \(6x - 5x + x + 2x = (6 - 5 + 1 + 2)x = 4x\).

Step 3: Simplify Constant Terms

Since there is only one constant term, \(-3\), it remains unchanged when combining like terms.

Step 4: Write the Simplified Expression

Combine the simplified variable term and the constant term to form the final expression: \(4x - 3\).

Key Concepts

Like Terms

In algebra, like terms are essential to simplifying and organizing expressions. Like terms are terms that have the same variable raised to the same power. They may differ in coefficients, which are the numerical part of the terms, but their variable components must be identical.
For instance, in the expression given, the terms involving the variable \(x\) — \(6x, -5x, x, \text{ and } 2x\) — are like terms because they share the base variable \(x\).
Recognizing like terms is a fundamental skill as it sets the stage for operations like addition and subtraction within an algebraic expression.

Combining Like Terms

Once you've identified like terms, the next step is to combine them. Combining like terms simplifies an expression and makes it easier to work with. To combine, you focus on the coefficients and perform arithmetic operations while keeping the variable part unchanged.
Using the original expression \(6x - 5x + x - 3 + 2x\), we first focus on the variable terms \(6x, -5x, x, \text{ and } 2x\). Add their coefficients:

• \(6 - 5 + 1 + 2 = 4\)

This results in the term \(4x\).
Combining like terms reduces the complexity of the expression, making further calculations straightforward and efficient.

Variable Terms

Variable terms are components of an algebraic expression that contain variables combined with coefficients. A variable is a symbol, often \(x\), \(y\), or \(z\), which represents a number that can vary. The coefficient is a number that is multiplied by the variable.
In our expression, the variable terms are \(6x, -5x, x, \text{ and } 2x\). Each of these terms contains \(x\) as the variable, with respective coefficients of 6, -5, 1, and 2.
Understanding variable terms is crucial as they represent the unknown in equations, allowing us to express mathematical relationships and solve problems involving variables.

Constant Terms

Constant terms in an algebraic expression are terms without a variable. They are fixed values that do not change. Constant terms stand alone and are simply numbers.

• Examples include numbers like 2, -3, 0.5, etc.

In the expression at hand, \(-3\) is the constant term. Since constant terms are not affected by the variables, when you combine like terms, this part of the expression remains unchanged, retaining its identity within the simplified expression.
Constant terms help in providing the baseline when evaluating expressions and equations, making them indispensable components of algebra.