Problem 11 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$15 x-4+6 x-9$$

Step-by-Step Solution

Answer

The simplified expression is \(21x - 13\).

Step 1: Identify Similar Terms

Identify the terms in the expression that have the same variable factor. Here we have two terms with the variable factor x: \(15x\) and \(6x\). There are no other terms with variables in the expression, so these are the like terms we can combine.

Step 2: Combine Similar Terms

Add the coefficients of like terms together. The coefficients of \(x\) are 15 and 6. Combining these gives \(15x + 6x = 21x\).

Step 3: Simplify Constant Terms

Identify constant terms in the expression, which are \(-4\) and \(-9\). Combine these by adding: \(-4 + (-9) = -13\).

Step 4: Write the Simplified Expression

Combine the simplified like terms and the constant terms to form the final simplified expression. Thus, the simplified expression is \(21x - 13\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is an essential skill that helps simplify expressions. Like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients. For instance, consider the expression:

• \(15x + 6x\) is an example, where both terms have the variable \(x\) raised to the first power. This means they are like terms.

Combining these, the task becomes adding the coefficients of these terms:

• \(15 + 6 = 21\), so \(15x + 6x = 21x\).

This principle significantly reduces the complexity of algebraic expressions and facilitates easy manipulation of equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that define a specific set of values. They often involve:

• Variables like \(x\) and \(y\), which represent unknowns or quantities that can vary.
• Constants and coefficients, such as \(4\), \(15x\), which are fixed values multiplying the variables.
• Operations including addition, subtraction, multiplication, and division.

For example, the expression \(15x - 4 + 6x - 9\) is an algebraic expression, comprising both variable terms and constants. Simplifying algebraic expressions often involves combining like terms and managing constant terms, as illustrated in the exercise above.

Constants in Algebra

Constants are standalone numbers in an algebraic expression without any variables attached. These values remain fixed and are crucial when simplifying expressions. In the expression \(15x - 4 + 6x - 9\), the constants are

• \(-4\)
• \(-9\)

Simplifying constants involves straightforward arithmetic. For example, adding the constants in this expression

• \(-4 + (-9) = -13\)

Incorporating these simplified constants into an expression is the final step, as it renders the expression as simple as possible, resulting in \(21x - 13\). Constants' roles may appear minor, but they have a significant impact on the overall value of the expression.