Problem 55 - Algebra 2

Question

Simplify each expression. 6\((2 x+3 y-1)\)

Step-by-Step Solution

Answer

12x + 18y - 6

Step 1: Distribute the 6

The expression is 6(2x + 3y - 1). To simplify the expression, start by distributing the 6 to each term inside the parentheses. This means you multiply each term inside the parentheses by 6. You will perform the following multiplications: 1. 6 * 2x = 12x 2. 6 * 3y = 18y 3. 6 * -1 = -6.

Step 2: Combine the results

After distributing the 6, you combine the results from the multiplications to write the simplified expression: Combine 12x, 18y, and -6 together to form the expression: 12x + 18y - 6.

Key Concepts

Simplification

Simplification is a process used to make expressions easier to understand or manage. In the context of algebra, when we simplify a polynomial expression, we aim to make it less complicated without changing its value.
For this exercise, simplification means reducing: 6\((2x + 3y - 1)\) into a more straightforward form.
The simplification process involves combining like terms, reducing coefficients, and removing any grouping symbols such as parentheses by performing operations like distribution. The result of simplifying this expression steps us towards a clearer and more compact form: 12x + 18y - 6.

• Combining like terms when possible.
• Removing unnecessary parentheses.
• Reducing coefficients correctly.

Simplification not only helps in making expressions manageable but also ensures you can efficiently carry out further operations such as evaluation or solving equations.

Distribution

Distribution is a fundamental process in algebraic manipulation where we apply a value outside of a parenthesis to each term inside the parenthesis. This exercise demonstrates distribution as it shows us how to handle expressions like 6\((2x + 3y - 1)\).
The distribution process involves taking the 6 and multiplying it by each term within the parentheses:

• Multiply 6 by 2x to get 12x.
• Multiply 6 by 3y to get 18y.
• Multiply 6 by -1 to result in -6.

The distribution operation helps in expanding expressions and is essential for simplifying complex polynomials.
It ensures that every term is treated equally and correctly, maintaining the integrity of the original equation as we simplify it.

Algebraic Expressions

Algebraic expressions form the basis for many problems in algebra and are composed of variables, numbers, and operations. An expression such as 6\((2x + 3y - 1)\) includes these components, all arranged in a specific order to define a particular relationship or equation.
These expressions don't have an "equals" sign, distinguishing them from algebraic equations.
In working with algebraic expressions, such as the one in this exercise, we often perform operations like simplification and distribution to better understand or solve them.

• Variables represent unknown values and are critical in making expressions generalizable.
• Operations such as addition and multiplication are used to combine variables and constants.
• Simplification helps reduce the complexity of expressions while preserving their inherent meaning.

Mastery of algebraic expressions is crucial, as it allows students to develop skills necessary for solving equations and modeling real-world situations using algebra.