Problem 79 - Elementary Algebra

Question

\(\frac{1}{2}(6 c+9)-\frac{3}{4}(8 c-11)\)

Step-by-Step Solution

Answer

-3c + 12.75

Step 1 - Distribute the fractions

Distribute \( \frac{1}{2} \) to both terms inside the first parentheses and distribute \( \frac{3}{4} \) to both terms inside the second parentheses. \( \frac{1}{2}(6c) + \frac{1}{2}(9) - \frac{3}{4}(8c) + \frac{3}{4}(11) \)

Step 2 - Simplify each distributed term

Calculate the simplified expression for each term: \( \frac{1}{2} \times 6c = 3c \) \ \( \frac{1}{2} \times 9 = 4.5 \) \ \( \frac{3}{4} \times 8c = 6c \) \ \( \frac{3}{4} \times (-11) = -8.25 \)

Step 3 - Substitute simplified terms back

Combine the simplified terms into the expression: \( 3c + 4.5 - 6c + 8.25 \)

Step 4 - Combine like terms

Combine the terms with \( c \) and the constant terms: \( 3c - 6c + 4.5 + 8.25 = -3c + 12.75 \)

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra. It allows us to simplify expressions by distributing a multiplier across terms inside parentheses.
For example, in the expression \(a(b + c)\), we apply the distributive property to get \(ab + ac\).
The original exercise requires you to distribute the fractions \(\frac{1}{2}\) and \(\frac{3}{4}\) to the terms within each set of parentheses.
Breaking it down:

• First, distribute \(\frac{1}{2}\) across the terms inside the first parentheses: \(\frac{1}{2}(6c + 9) = \frac{1}{2} \cdot 6c + \frac{1}{2} \cdot 9 = 3c + 4.5\).
• Next, distribute \(\frac{3}{4}\) across the terms in the second parentheses: \(\frac{3}{4}(8c - 11) = \frac{3}{4} \cdot 8c - \frac{3}{4} \cdot 11 = 6c - 8.25\).

This step helps break down the expression into simpler parts, making it easier to manage.

Combining Like Terms

Combining like terms is another critical step when simplifying expressions. Like terms are terms that have the same variable raised to the same power.
For example, \(3c\) and \(6c\) are like terms because they both have the variable \(c\).
After using the distributive property in our exercise, we have the expression: \(3c + 4.5 - 6c + 8.25\).
We need to combine the like terms:

• Combine the \(c\) terms: \(3c - 6c = -3c\).
• Combine the constant terms: \(4.5 + 8.25 = 12.75\).

Combining these results gives us the simplified expression \(-3c + 12.75\).
Combining like terms reduces the expression to its simplest form.

Fractions in Algebra

Working with fractions in algebra can seem challenging, but the process is straightforward with practice. It often involves distributing fractions across terms and then simplifying.
For instance, distributing \(\frac{1}{2}\) over the terms inside parentheses in the original exercise:

• \(\frac{1}{2}(6c + 9) = \frac{1}{2} \cdot 6c + \frac{1}{2} \cdot 9 = 3c + 4.5\).

Similarly, distributing \(\frac{3}{4}\) over the terms inside parentheses:

• \(\frac{3}{4}(8c - 11) = \frac{3}{4} \cdot 8c - \frac{3}{4} \cdot 11 = 6c - 8.25\).

Once distributed, the fractions work just like whole numbers and make simplifying the expression easier.
This approach helps break down complex problems into manageable parts.

Simplifying Expressions

Simplifying expressions is the ultimate goal in many algebra problems. It involves several steps, including using the distributive property, combining like terms, and handling fractions.
In the original exercise, we simplified the expression step by step:

• We first distributed the fractions: \(\frac{1}{2}(6c + 9)\) and \(\frac{3}{4}(8c - 11)\).
• Then, we simplified each term: \(3c + 4.5 - 6c - 8.25\).
• Finally, we combined like terms to get the final simplified expression: \(-3c + 12.75\).

Each step of the process gets us closer to the simplest form of the expression, where no further simplification is possible.
This methodical approach is crucial to solving more complex algebraic expressions.