Problem 19 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \(3 a+4(2 a-3)\)

Step-by-Step Solution

Answer

The simplest form of the expression is \(11a - 12\).

Step 1: Distribute inside the parentheses

Start by distributing the 4 across the terms inside the parentheses: \(4(2a - 3)\) becomes \(4 \times 2a - 4 \times 3\). This simplifies to \(8a - 12\).

Step 2: Combine like terms

Now, add the term \(3a\) from the expression \(3a + 8a - 12\). Combining like terms gives: \((3a + 8a) - 12 = 11a - 12\).

Step 3: Write the expression in simplest form

Since there are no more like terms to combine and no common factors to factor out, the expression is already in its simplest form. Thus, \(11a - 12\) is the result.

Key Concepts

Understanding the Distributive Property

The distributive property is one of the most fundamental concepts in algebra. It's a handy tool that helps us to simplify expressions by removing parentheses and rewriting expressions in a different form. The distributive property states:

• For any numbers or variables, if you have an expression like \(a(b + c)\), you can distribute \(a\) to both \(b\) and \(c\).
• This expression becomes \(a \times b + a \times c\).

This means you're multiplying the term outside the parentheses by each term inside the parentheses individually. Let's see how this applies in our example:
We started with the expression \(4(2a - 3)\). By applying the distributive property, it becomes \(4 \times 2a - 4 \times 3\), which simplifies to \(8a - 12\).
Understanding this property helps with breaking down more complex expressions and is a skill you will use often in algebra!

Mastering the Art of Combining Like Terms

Once you've applied the distributive property, your next task is often to combine like terms. This is a crucial step for simplification in algebra.
Like terms are terms in an expression that contain the same variable raised to the same power. To combine them, you simply add or subtract their coefficients, which are the numbers in front of the variables.

• In our example, the terms \(3a\) and \(8a\) both contain the variable \(a\).
• This means they are like terms and can be combined. Adding the coefficients gives us \(11a\).

Now the expression has been reduced to \(11a - 12\).
Combining like terms effectively simplifies the expression further, making it easier to work with later.

Using Algebraic Operations to Simplify Expressions

Algebraic operations refer to the basic arithmetic operations we use to manipulate algebraic expressions: addition, subtraction, multiplication, and division.
These operations allow us to manipulate and simplify expressions and are foundational skills in algebra.

• In our original problem, we used multiplication to apply the distributive property.
• Then, we used addition to combine like terms.
• Finally, we rewrite the expression in its simplest form.

Each step involved an algebraic operation to bring clarity and simplicity to the expression.
In doing so, the expression \(3a + 4(2a - 3)\) was simplified to \(11a - 12\).
Mastering these operations is key to becoming proficient in algebra, as they allow you to solve and simplify problems efficiently.