Problem 106 - Intermediate Algebra

Question

Simplify each expression. $$ -7(a-3)-5[3(a-4)-2(a+2)] $$

Step-by-Step Solution

Answer

The expression simplifies to \(-12a + 101\).

Step 1: Distribute within the first parenthesis

Simplify the expression \(-7(a-3)\) by distributing \(-7\) through the terms in the parenthesis. This gives: \[-7 \cdot a + 7 \cdot 3 = -7a + 21.\]

Step 2: Distribute within the inner bracket

Simplify \(3(a-4)\) by distributing \(3\) to each term inside: \[3 \cdot a - 3 \cdot 4 = 3a - 12.\]Next, simplify \(-2(a+2)\) by distributing \(-2\) as follows: \[-2 \cdot a - 2 \cdot 2 = -2a - 4.\]

Step 3: Combine terms in the bracket

Combine \(3a - 12\) and \(-2a - 4\) inside the brackets:\[(3a - 12) - (2a + 4) = 3a - 12 - 2a - 4 = (3a - 2a) + (-12 - 4) = a - 16.\]

Step 4: Distribute the negative number through the bracket

Distribute \(-5\) through \((a - 16)\):\[-5 \cdot a - 5 \cdot (-16) = -5a + 80.\]

Step 5: Combine like terms

Now add together all distributed terms:\((-7a + 21) - (5a - 80)\) becomes:\[-7a + 21 - 5a + 80 = (-7a - 5a) + (21 + 80) = -12a + 101.\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that involves multiplying a single term by each term inside a parenthesis. It's often written as \( a(b + c) = ab + ac \), which means you distribute (or multiply) the term outside the parenthesis to each of the terms within.

For example, in our expression

• \(-7(a-3)\), we distribute \(-7\) across \(a\) and \(-3\), giving us \(-7a + 21\).
• Similarly, to simplify \(3(a-4)\), we distribute \(3\) to both terms inside, resulting in \(3a - 12\).

The distributive property is invaluable for handling complex expressions where simplification is necessary. It allows us to eliminate brackets and combine terms later in the solution process. Remember to carefully distribute negative signs; this maintains the correct signs for each term during simplification.

Combining Like Terms

Combining like terms is an essential skill when simplifying algebraic expressions. Like terms are terms in an expression that have the same variable raised to the same power. Those terms can be easily added together by summing their coefficients.

Here's how it works in our problem:

• In the step \[(3a - 12) - (2a + 4),\]
we combine \(3a\) and \(-2a\) because they both contain the variable \(a\). This results in \((3 - 2)a = a\).
• Similarly, we combine \(-12\) and \(-4\), both constant terms, to get \(-16\).

By the end of this step, any terms that can be combined are combined, simplifying the expression further. When simplifying, always ensure you maintain the correct order of operations, as it can impact how terms are combined.

Order of Operations

The order of operations is a guiding principle in mathematics that dictates the sequence in which the operations should be executed to guarantee accurate results. A helpful acronym to remember is PEMDAS, which stands for:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M/D: Multiplication and Division (left-to-right)
• A/S: Addition and Subtraction (left-to-right)

In our exercise, this principle ensures that we:

• Simplify expressions inside parentheses first, using the distributive property where necessary.
• Combine like terms next.
• Handle any operations outside of parentheses, ensuring multiplication or division precedes addition or subtraction.

For example, after we distribute and combine like terms, we follow the order by carrying out operations in the correct sequence, leading to the final solution. Remember, attention to the order of operations prevents mistakes and ensures accuracy when simplifying expressions.