Problem 86 - Algebra A Combined Function

Question

Simplify each expression. See Section \(1.8 .\) \(-(3 a-3)+2 a-6\)

Step-by-Step Solution

Answer

The simplified expression is \(-a - 3\).

Step 1: Distribute the Negative Sign

The expression is equation\[ -(3a - 3) + 2a - 6 \] inside the parentheses. To distribute the negative sign, multiply each term inside the parentheses by -1, resulting in\[ -3a + 3 + 2a - 6 \] This changes both the sign of the terms inside the parentheses.

Step 2: Combine Like Terms for 'a'

Identify the terms that contain the variable \(a\). These terms are \(-3a\) and \(2a\).Combine them by adding their coefficients:\[ -3a + 2a = -1a \]which simplifies to \[ -a \].So now the expression becomes:\[ -a + 3 - 6 \]

Step 3: Combine Constant Terms

Next, deal with the constant terms. The constants are 3 and -6.Compute their sum:\[ 3 - 6 = -3 \]Now, the expression simplifies to:\[ -a - 3 \]

Step 4: Final Simplified Expression

The simplified version of the original expression \[-(3a - 3) + 2a - 6 \]is\[-a - 3 \]where all like terms have been combined, and the expression is now in its simplest form.

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra that helps in simplifying expressions, especially when dealing with parentheses. When you see a negative sign or a number before parentheses, it indicates you need to distribute that factor to each term inside the parentheses. Imagine wrapping a gift; you need to ensure every part of the gift is covered. In the expression \[ -(3a - 3) + 2a - 6 \] the negative sign before the parentheses means we multiply each term inside the parentheses by \(-1\). This changes the signs of the terms, resulting in:- The term \(3a\) becomes \(-3a\)- The term \(-3\) becomes \(+3\)So after applying the distributive property, the expression turns into:\[ -3a + 3 + 2a - 6 \]This step sets the stage for further simplification by changing the signs of the terms inside the parentheses as necessary.

Combining Like Terms

After distributing the negative sign, the expression becomes:\[-3a + 3 + 2a - 6 \]. The next step involves combining like terms. Like terms are terms whose variables and exponents are the same. Here, the like terms are the ones involving the variable \(a\):- The terms \(-3a\) and \(2a\) both contain the variable \(a\).To combine them, we add their coefficients together: \(-3 + 2 = -1\). This gives us:\[-1a\] or simply \(-a\).Now, the expression simplifies further to:\[-a + 3 - 6\].Combining like terms is pivotal in algebra as it helps condense an expression, making it easier to interpret and solve. Always align similar variable terms before proceeding with other calculations.

Constant Terms

Once we have combined the terms with variables, we focus on the constant terms. Constants are numbers without attached variables. In the expression \[-a + 3 - 6\], the constants are \(+3\) and \(-6\).- Addition or subtraction of constants is straightforward: \(3 - 6 = -3\)Thus, the expression transforms into:\[-a - 3\].Simplifying these constants ensures that there are no unnecessary numbers floating in the expression. Keeping track of these constants allows you to resolve remaining simple numeric calculations and achieve the most reduced form of the expression.