Problem 107 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 6-4(-3 c-7) $$

Step-by-Step Solution

Answer

The simplified expression is \(12c + 34\).

Step 1: Distribute the Negative Sign

To simplify the expression, start by distributing the -4 across the terms inside the parentheses. Multiply -4 by each of the terms within the parentheses, \(-3c\) and \(-7\). This gives us:\[-4(-3c) + (-4)(-7)\].

Step 2: Simplify Each Multiplied Term

Now, multiply -4 by -3c which results in \(12c\), and multiply -4 by -7 which results in \(28\). Therefore, the expression becomes:\[6 + 12c + 28\].

Step 3: Combine Like Terms

Next, combine the like terms. The constants 6 and 28 can be added together to simplify the expression further. Add these two numbers:\[6 + 28 = 34\].Thus, the simplified expression is:\[12c + 34\].

Key Concepts

Distributive Property

The Distributive Property is a fundamental algebraic principle. It helps us multiply a single term by each term inside a parenthesis. Imagine it as a way to "unlock" terms from parentheses so everyone gets multiplied.

In mathematics, the Distributive Property is framed as:
\( a(b + c) = ab + ac \).

This means you take the term outside the parentheses, like \(-4\) in our example, and multiply it by each term inside.

• In the expression \(-4(-3c-7)\), the \(-4\) multiplies both \(-3c\) and \(-7\).
• This operation gives us: \( -4(-3c) + (-4)(-7) \).

By applying the Distributive Property correctly, we transform expressions into simpler forms, ready to be further simplified.

Combining Like Terms

Combining like terms is like grouping similar actors on a stage. In algebra, terms with the same variable parts can be added or subtracted.

Like terms have the same variable raised to the same power, while different terms have different variables or degrees. For example, \(12c\) and \(34\) are unlike terms because one has a variable and the other does not.

In our expression, \(6 + 12c + 28\):

• \(6\) and \(28\) are constants, hence like terms. They can be combined to \(34\).
• \(12c\) stands alone because it has a different form.

By grouping and combining like terms, you simplify the expression to its most compact form, which in this case is \(12c + 34\).

Negative Sign Distribution

Distributing a negative sign is like using a flashlight to shine on everything hidden in the parentheses. It ensures each term gets the appropriate sign.

When distributing a negative sign, multiply by \(-1\), creating a mirror effect on each term's sign. Let's explore this principle using \(-4(-3c - 7)\):

• A negative times a negative, \(-4\) and \(-3c\), results in a positive: \(12c\).
• A negative times a negative, \(-4\) and \(-7\), also results in a positive: \(28\).

Each term in the parentheses changes sign when a negative number distributes over them. This strategic distribution allows for a correct simplification of complex expressions.