Problem 31 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ -(2 a-b-6 c) $$

Step-by-Step Solution

Answer

-2a + b + 6c

Step 1: Distribute the Negative Sign

Distribute the negative sign across each term within the parentheses.

Step 2: Apply the Distribution

Multiply each term inside the parentheses by -1: -1 * (2a) = -2a -1 * (-b) = b -1 * (-6c) = 6c

Step 3: Combine the Results

Combine the results from the distribution step to get: -2a + b + 6c.

Key Concepts

Distributive Property

In algebra, the distributive property helps to simplify expressions by removing parentheses. The property essentially states that a term outside the parentheses must be multiplied with each term inside the parentheses. If you have an expression of the form \[ a(b + c) \], you can distribute 'a' across 'b' and 'c' to get \[ ab + ac \].
This technique is useful in breaking down complex expressions into simpler terms, helping you manage and simplify calculations more easily. Always ensure to distribute the term completely, covering all terms inside the parentheses.

Negative Sign Distribution

When distributing a negative sign, you're essentially multiplying each term inside the parentheses by -1. This changes the sign of each term. For instance, given \[ -(2a - b - 6c) \], distribute the -1 across the terms:
\[-1 \times (2a) = -2a \],
\[-1 \times (-b) = b \],
\[-1 \times (-6c) = 6c \].
The resulting expression will be \[ -2a + b + 6c \]. Always remember that distributing a negative sign reverses the signs within the parentheses, turning each positive term into a negative one and vice versa.

Combining Like Terms

Combining like terms simplifies an expression by aggregating terms with the same variable components. For example, in the expression \[ -2a + b + 6c \], no terms have the same variable, so there are no like terms to combine.
However, combining like terms becomes crucial when you have multiple terms with identical variables. For example, in \[ 3a - 2a + 4b + b \], we can combine \[ 3a - 2a = a \] and \[ 4b + b = 5b \], resulting in \[ a + 5b \].
This process makes complex expressions more straightforward and significantly easier to solve. Always ensure to bring together coefficients of the same variable and perform the necessary arithmetic.