Problem 26 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$3(2 x-3)-7(3 x-1)$$

Step-by-Step Solution

Answer

The simplified expression is \(-15x - 2\).

Step 1: Distribute First Term

Multiply each term inside the first set of parentheses by 3. This gives: \[3(2x) - 3(3) = 6x - 9\]

Step 2: Distribute Second Term

Multiply each term inside the second set of parentheses by -7. This gives:\[-7(3x) + 7(1) = -21x + 7\]

Step 3: Combine All Terms

Bring together all the terms from both distributions: \[6x - 9 - 21x + 7\]

Step 4: Combine Similar Terms

Group and combine the like terms:\[(6x - 21x) + (-9 + 7) = -15x - 2\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions and solving equations. It's particularly handy when you need to remove parentheses in algebraic expressions. In simpler terms, this property allows you to multiply a single term outside the parentheses by each term inside the parentheses. For instance, look at the expression given in the exercise: \(3(2x - 3)\). To simplify this using the distributive property, you perform the following steps:

• Multiply the 3 by every term inside the parentheses.
• This results in: \(3 \cdot 2x = 6x\) and \(3 \cdot -3 = -9\).

Applying the distributive property helps to transform complex expressions into simpler ones that are easier to work with. It's crucial to remember that you must apply the multiplication to each term inside the parentheses separately.

Combining Like Terms

Once you have applied the distributive property to an expression, the next step is typically to combine like terms. Like terms are terms that contain the same variable raised to the same power. For example, in the expression \(6x - 9 - 21x + 7\), the like terms are \(6x\) and \(-21x\), as they both contain the variable \(x\). Here's how you combine the like terms in our example:

• Identify the like terms which are the coefficients of \(x\) in this case.
• Add or subtract the coefficients: \(6x - 21x = -15x\).
• Then, combine the constant terms: \(-9 + 7 = -2\).

By combining like terms, you reduce the expression into a simpler form, making it easier to handle in equations or further calculations.

Algebraic Manipulation

Algebraic manipulation involves various techniques employed to transform and simplify algebraic expressions. It serves as a toolkit to handle expressions more efficiently. In this problem, algebraic manipulation starts with distributing, as mentioned previously, and then proceeds to combining like terms.The ultimate goal is to rearrange the expression into its simplest form, which is often essential for solving equations or understanding relationships between variables.

• Ensure each step logically follows from the previous one—first distribute, then combine.
• Double-check that like terms are correctly grouped and simplified.
• Keep the expressions neat to avoid mistakes; the final simplified form of our example is \(-15x - 2\).

Mastery of algebraic manipulation provides a solid foundation for tackling more advanced topics in mathematics, bridging basic operations to complex problem-solving scenarios.