Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (2 x+3)+(4 x-6) $$

Step-by-Step Solution

Answer

The simplified expression is \(6x - 3\).

Step 1: Distribute the addition operation

We can distribute the addition operation to each term inside both expressions like this: \[ (2x+3)+(4x-6) = 2x+3+4x-6 \] Now our expressions are combined into a single expression.

Step 2: Combine like terms

Identify and group the terms involving x and the constant terms for simplification: \[ 2x+4x + (3-6) \]

Step 3: Simplify the grouped terms

Calculate the coefficients of the x-terms and simplify the constants: \[ (2+4)x + (-3) = 6x - 3 \] The simplified expression is: \(6x - 3\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental process for simplifying expressions. Like terms are terms that have the same variables raised to the same powers, even if their coefficients (the numbers in front of the variables) are different. For instance, in the expression \(2x + 4x\), both terms are like terms because they both contain the single variable \(x\) raised to the first power.

To combine these like terms, you simply add or subtract the coefficients while keeping the variable part unchanged. Following our example, \(2x + 4x\) combines to \(6x\) because \(2 + 4 = 6\), and the \(x\) remains the same. This process helps in reducing the complexity of algebraic expressions and makes them more manageable and easier to understand.

Key Points in Combining Like Terms:

• Add or subtract the coefficients of like terms.
• Keep the variable and the power of the variable the same.
• Combining like terms comes after distributing any operations over the terms within parentheses.

Remember, terms with different variables or powers cannot be combined through this process.

Distributive Property

The distributive property is an algebraic rule that is essential for expanding and simplifying expressions. It allows you to multiply a single term by each term within a parenthesis, distributing the multiplication over addition or subtraction.

In the expression \((2x + 3) + (4x - 6)\), there's nothing to distribute in the traditional sense because we are not multiplying the terms by a value outside the parentheses. Instead, what we do is distribute the addition operation to remove the parentheses as shown in the step-by-step solution: \(2x + 3 + 4x - 6\). This sets us up for combining like terms.

Understanding the Distributive Property:

• It is written as \(a(b + c) = ab + ac\) or \(a(b - c) = ab - ac\).
• Use it to expand expressions and remove parentheses.
• It simplifies the process of combining like terms by making each term explicit.

The distributive property is a powerful tool that simplifies expressions and solves equations more efficiently.

Algebraic Operations

Algebraic operations encompass the basic arithmetic operations—addition, subtraction, multiplication, and division—when applied to algebraic expressions that may include numbers, variables, and exponents. Simplifying an algebraic expression often requires carrying out these operations in a specific order, following the rules of arithmetic and algebra.

For the given exercise, we use addition to remove parentheses and then combine like terms. The solution represents a sequential application of the fundamental algebraic operations:

1. Addition: We add the terms inside the parentheses.
2. Combination: We sum up the coefficients of the like terms, such as \(2x + 4x\) to get \(6x\).
3. Subtraction: We also simplify the constants by performing \(3 - 6\) to get \(-3\).

In conclusion, algebraic expressions are simplified by systematically applying algebraic operations, ensuring each step is performed accurately to reach a simplified expression, which in this case is \(6x - 3\).