Problem 2 - Intermediate Algebra

Question

Simplify the algebraic expressions by combining similar terms. $$ 5 x-8 x+x $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x\).

Step 1: Identify Similar Terms

First, analyze the expression to identify similar terms. Similar terms are terms that have the same variable raised to the same power. In the expression \(5x - 8x + x\), all terms are similar because they all contain the variable \(x\).

Step 2: Group the Similar Terms Together

Combine all the similar terms identified in the previous step. For \(5x - 8x + x\), group the coefficients of \(x\): \((5 - 8 + 1)x\).

Step 3: Perform the Arithmetic Operations

Calculate the sum of the coefficients of \(x\) obtained in the previous step: \(5 - 8 + 1 = -2\).

Step 4: Write the Simplified Expression

Incorporate the result from the arithmetic operation back into the expression: \(-2x\).

Key Concepts

Combining Like Terms

In algebra, simplifying expressions by combining like terms is a crucial skill. Like terms share the same variables and each variable is raised to the same power. This means they can be combined through addition or subtraction.

• Look for terms with identical variables.
• Even if their coefficients differ, ensure the variable portions match exactly.
• Only change the coefficients when combining like terms.

In the given example, all parts of the expression, namely \(5x\), \(-8x\), and \(x\), are like terms as they all feature the variable \(x\) raised to the first power. By combining these terms, the expression is greatly streamlined, resulting in a simpler equation to work with.

Simplification Techniques

Simplifying algebraic expressions makes them easier to interpret and work with. After identifying like terms, grouping them efficiently is the key to simplification.

• Group the coefficients of the like terms.
• Perform any necessary arithmetic operations.
• Simplify the resulting expression for clarity.

Using our example, once like terms \(5x\), \(-8x\), and \(x\) are identified, their coefficients \(5\), \(-8\), and \(1\) are grouped together. Adding these coefficients, which requires simple arithmetic, gives us \(-2\). Multiply this sum by the common variable \(x\) to achieve the simplified expression \(-2x\).

Variable Coefficients

Variable coefficients represent numbers that are multiplied by variables in an algebraic expression. Understanding these is crucial for combining like terms and simplifying expressions.

• Coefficients can be positive or negative.
• Arithmetic operations on coefficients lead to simpler expressions.
• The coefficient dictates the magnitude and direction of the term's value.

In the expression \(5x - 8x + x\), the coefficients are \(5\), \(-8\), and \(1\). By focusing arithmetic operations on these coefficients, we ensure the integrity of the expression while simplifying it. Thus, when these coefficients are combined to result in \(-2x\), the expression becomes more streamlined and easier to handle.