Problem 76 - Algebra 2

Question

Simplify each expression. $$ 4(2 a+5 b)-3(4 b-a) $$

Step-by-Step Solution

Answer

The simplified expression is \(11a + 8b\).

Step 1: Expand Each Term

Start by expanding the terms inside the parentheses. The expression is given as \[ 4(2a+5b) - 3(4b-a) \]First, expand each term:1. \(4(2a + 5b) = 4 \cdot 2a + 4 \cdot 5b = 8a + 20b\)2. \(-3(4b - a) = -3 \cdot 4b + (-3)(-a) = -12b + 3a\)So the expanded expression becomes \(8a + 20b - 12b + 3a\).

Step 2: Combine Like Terms

Combine the like terms from the expanded expression. We have terms with \(a\) and terms with \(b\):- Combine terms with \(a\):\(8a + 3a = 11a\)- Combine terms with \(b\):\(20b - 12b = 8b\)Thus, the expression simplifies to \(11a + 8b\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic property that allows you to multiply a single term across a sum or difference within parentheses. This property is expressed as \(a(b+c) = ab+ac\). It simplifies the process of dealing with expressions involving parentheses, enabling you to "distribute" the outer number across each term inside. In our original exercise, **4(2a+5b)-3(4b-a)**, we applied the distributive property.

• First, multiply 4 by each term inside the first set of parentheses: \(4(2a+5b) = 8a + 20b\).
• Next, apply the property to the second part, noting the negative sign affects both terms: \(-3(4b-a) = -12b + 3a\).

The distributive property helps break down complex expressions into simpler parts, making them more manageable and easier to evaluate.

Combining Like Terms

Combining like terms is a process that helps simplify algebraic expressions by summing or subtracting coefficients of identical variable terms. The key is to look for terms that contain the same variable raised to the same power.In the expression **8a + 20b - 12b + 3a** derived from our exercise, you identify like terms:

• Terms involving \(a\): \(8a\) and \(3a\).
• Terms involving \(b\): \(20b\) and \(-12b\).

You then combine these terms:

• For \(a\): \(8a + 3a = 11a\).
• For \(b\): \(20b - 12b = 8b\).

After combining, the expression becomes \(11a + 8b\). This step simplifies the expression by reducing it to fewer terms, making it easier to handle especially for future calculations or equations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They are used to represent mathematical relationships or functions in a symbolic form and are the building blocks in algebra.In the exercise, the expression **4(2a+5b) - 3(4b-a)** is an algebraic expression. It includes:

• Variables: \(a\) and \(b\), representing unknown values in the expression.
• Arithmetic operations: addition, subtraction, and multiplication.
• Constants: numerical coefficients like 4 and 3, which multiply the variables.

Understanding algebraic expressions is fundamental as they serve as the foundation for constructing and solving equations. Simplifying these expressions involves using properties like the distributive property and techniques like combining like terms, to reduce them to their simplest form. This process makes it easier to analyze and solve mathematical problems involving unknown quantities.