Problem 26 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Simplified expression: \(-15x - 2\).

Step 1: Distribute the 3 across the terms in the parentheses

To begin, distribute the 3 to each term inside the first set of parentheses: \(3(2x - 3)\). This gives us \(3 \times 2x - 3 \times 3 = 6x - 9\).

Step 2: Distribute the -7 across the terms in the parentheses

Next, distribute the -7 to each term inside the second set of parentheses: \(-7(3x - 1)\). This results in \(-7 \times 3x - (-7) \times 1 = -21x + 7\).

Step 3: Combine the expressions

Now, combine the results from steps 1 and 2: \(6x - 9 - 21x + 7\).

Step 4: Combine like terms

Identify and combine like terms. For the variables: \(6x - 21x = -15x\). For the constants: \(-9 + 7 = -2\).

Step 5: Write the final simplified expression

The simplified expression after combining like terms is \(-15x - 2\).

Key Concepts

Distributive Property

The distributive property is an essential tool in simplifying algebraic expressions. It allows us to eliminate parentheses by distributing the multiplication over addition or subtraction inside the parentheses. For example, in the expression \(3(2x - 3)\), we apply the distributive property by multiplying 3 with each term inside the parentheses. This operation gives us \(3 \times 2x\) and \(3 \times -3\), resulting in the expression \(6x - 9\).
This property not only helps in simplifying but also in understanding how different parts of an expression relate based on their coefficients.
When executing the distributive property:

• Apply the multiplication to each term inside the parentheses separately.
• Be careful with negative signs; multiplying a negative number also distributes that negativity.

Understanding and correctly using the distributive property prepares students to tackle more complicated algebraic problems effectively.

Combining Like Terms

Once we have distributed the multiplication across the terms inside the parentheses, the next step is to combine like terms. "Like terms" are terms that have the same variable part and can thus be added together. In the expression from the step-by-step solution, \(6x - 9 - 21x + 7\), we group the terms with the variable \(x\) (i.e., \(6x\) and \(-21x\)) and the constant terms (i.e., \(-9\) and \(7\)).

• For the terms with \(x\): \(6x\) is like \(-21x\), so we sum them to get \(-15x\).
• For the constant terms: \(-9\) and \(7\) combine to form \(-2\).

It's important to ensure that only similar terms are combined. Variable terms must share the same variable and exponent, while constants are combined based on their numerical values. Mastering this step aids in achieving a simplified expression as the final output.

Algebraic Simplification

Algebraic simplification is the process used to make an expression more compact and manageable, bring clarity, and often make further calculations easier. Simplifying an algebraic expression involves removing parentheses, using the distributive property, and combining like terms to rewrite the expression in its simplest form.
In the problem provided, after appropriately using these steps, we finish with the expression \(-15x - 2\). This signifies we have removed all parentheses and brought similar terms together.
Benefits of algebraic simplification:

• It reduces the complexity of expressions.
• Helps in identifying trends and solutions more efficiently in algebra problems.
• Makes equations that involve the simplified expression easier to solve or interpret.

By practicing and understanding algebraic simplification, students can enhance their problem-solving skills and better handle complex algebraic expressions.