Problem 28 - Elementary Algebra

Question

Simplify. $$ \left(2 y_{2}+6 y-8\right)-\left(5 y_{2}-12 y+1\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3y_{2} + 18y - 9\).

Step 1: Distribute the Negative Sign

You need to distribute the negative sign across the second group of terms in the expression. The expression becomes: \[ (2y_{2} + 6y - 8) - 1(5y_{2} - 12y + 1) = 2y_{2} + 6y - 8 - 5y_{2} + 12y - 1 \]

Step 2: Combine Like Terms for \(y_{2}\)

Identify and combine like terms for \(y_{2}\). The terms are \(2y_{2}\) and \(-5y_{2}\). \[ 2y_{2} - 5y_{2} = -3y_{2} \]

Step 3: Combine Like Terms for \(y\)

Identify and combine like terms for \(y\). The terms are \(6y\) and \(12y\). \[ 6y + 12y = 18y \]

Step 4: Combine Constant Terms

Combine the constant terms \(-8\) and \(-1\). \[ -8 - 1 = -9 \]

Step 5: Write the Final Simplified Expression

The simplified result is obtained by putting together all the results from the previous steps: \[ -3y_{2} + 18y - 9 \]

Key Concepts

Distributive Property

The distributive property is a key tool in algebra, allowing us to simplify expressions more efficiently. This principle tells us that when we have an expression like \( a(b + c) \), it can be expanded to \( ab + ac \). This process is called

• Distribution: Apply multiplication to each term inside the parenthesis with the term outside them.
• Sign consideration: While distributing, it's crucial to mind the signs of all terms, especially when dealing with subtraction.

In the given problem, we employ the distributive property to handle the subtraction correctly. Specifically, we must distribute the negative sign to each term inside the parenthesis of the second expression. This transforms the expression into: \[2y_{2} + 6y - 8 - 5y_{2} + 12y - 1\] This step is crucial to ensuring that each term is appropriately adjusted for addition and subtraction later in the process.

Combining Like Terms

Combining like terms is one of the fundamental steps in simplifying algebraic expressions. It involves:

• Identifying similar terms: Like terms have identical variable parts (e.g., both have the same variables raised to the same powers).
• Sum or subtract the coefficients: Only the numerical coefficients of the like terms are added or subtracted, not the variables themselves.

Let's look at the exercise. After distributing the negative sign, our job is to combine terms:- For the terms with \( y_2 \), we have \( 2y_2 \) and \( -5y_{2} \), which combine to form \( -3y_{2} \).- Similarly, the terms with \( y \) are \( 6y \) and \( +12y \), totaling \( 18y \).- Finally, the constant terms \(-8\) and \(-1\) combine to \(-9\).This operation significantly reduces the complexity of the expression, making it much easier to manage and understand.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial to its simplest form. This enables easier evaluation and manipulation in further mathematical operations. In simplifying polynomials, the core steps include:

• Using the distributive property to remove parentheses.
• Combining like terms to collapse the expression into fewer, simpler terms.

In our exercise, the original polynomial expression was simplified from:\[(2y_{2} + 6y - 8) - (5y_{2} - 12y + 1)\]into a more concise form:\[-3y_{2} + 18y - 9\]This expression retains all the essential algebraic information of the initial polynomial but is neater and more practical for computation. Simplifying expressions not only helps in solving equations but also improves clarity when communicating mathematical ideas.