Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$-10 x+2 x$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \( -8x \).

Step 1: Identify common factors

In the given expression, \( -10x + 2x \), there are two terms, both of which are multiples of 'x'. Therefore, 'x' is the common factor.

Step 2: Use the addition property

Add the numerical coefficients of the like terms in the expression. It's important to keep the sign of each term in mind while adding. In this case, \( -10 + 2 = -8 \).

Step 3: Final simplification

The simplified form of the expression will be the result from step 2 multiplied by the common factor. In this case, the common factor is 'x', so the simplified expression is \( -8x \).

Key Concepts

Common Factors

When simplifying algebraic expressions, spotting common factors is a crucial step. A common factor is an expression, number, or variable that divides each term in a sequence without a remainder. For instance, in the expression \(-10x + 2x\), each term contains 'x' as a factor.
By identifying 'x' as the common factor, we can rewrite the original expression and factor it out. Doing so helps simplify calculations and transforms the expression to a more simplified form. Understanding common factors ensures the simplification process is efficient and precise.
Remember:

• Identify variables or numbers that appear in each term.
• Factor these out of the expression.

This simplifies the problem and prepares it for further reduction.

Coefficient Addition

Addition of coefficients is the next step once common factors are identified. Coefficients are the numerical components attached to variables in an expression. In the expression \(-10x + 2x\), \(-10\) and \(2\) are the coefficients associated with the variable 'x'.
To simplify, add these coefficients together. Keep an eye on the signs, as negative and positive numbers affect the sum:

• Identify each coefficient and note its sign.
• Add the coefficients: \(-10 + 2 = -8\).

This results in a new coefficient that accompanies the common factor. Proper addition of coefficients aligns the algebraic expression towards its most reduced form while maintaining accuracy.

Like Terms

In algebra, like terms are terms that have identical variables raised to the same power. Identifying like terms is crucial for simplifying expressions because they can be combined in calculations. For example, \(-10x\) and \(2x\) are like terms because they both contain the variable 'x' to the first power.
By consolidating like terms, the expression becomes easier to work with. It allows for the straightforward addition or subtraction of coefficients to reach a simpler expression.
Remember:

• Check variable and exponent consistency to determine like terms.
• Once terms are consolidated, use operations like addition and subtraction on their coefficients.

Combining like terms is a fundamental process in algebra that leads to cleaner, more understandable expressions.