Problem 24 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 5(3 x-1)+6(2 x+3) $$

Step-by-Step Solution

Answer

The simplified expression is \(27x + 13\).

Step 1: Distribute the first term

Start by distributing the 5 in the expression \(5(3x - 1)\). Multiply 5 by each term inside the parentheses: \(5 \times 3x = 15x\) and \(5 \times (-1) = -5\). Thus, \(5(3x - 1) = 15x - 5\).

Step 2: Distribute the second term

Now distribute the 6 in the expression \(6(2x + 3)\). Multiply 6 by each term inside the parentheses: \(6 \times 2x = 12x\) and \(6 \times 3 = 18\). Thus, \(6(2x + 3) = 12x + 18\).

Step 3: Combine distributed expressions

With the results from the previous steps, combine the expressions: \(15x - 5\) and \(12x + 18\). This gives us: \(15x - 5 + 12x + 18\).

Step 4: Combine like terms

Combine like terms by adding coefficients of the same variable and summing the constants: \(15x + 12x = 27x\) and \(-5 + 18 = 13\). Therefore, the simplified expression is \(27x + 13\).

Key Concepts

Distributive Property

The distributive property is a handy tool in algebra that helps simplify expressions and make calculations easier. It's particularly useful when you need to get rid of parentheses by multiplying the term outside of them across each element inside.

Imagine you have an expression like this: \(5(3x - 1)\). According to the distributive property, you multiply 5 by each term inside the parentheses. So, you multiply 5 by \(3x\) to get \(15x\), and 5 by \(-1\) to get \(-5\).

• Visual: \(5 \times 3x = 15x\)
• Visual: \(5 \times (-1) = -5\)

By distributing, we simplify the expression to \(15x - 5\).

This property can be applied anytime you have a number outside the parentheses, and it ensures that every term inside gets multiplied. This makes subsequent operations a lot more manageable.

Combining Like Terms

Combining like terms is essentially adding or subtracting terms that have the same variables raised to the same power. It's a crucial skill to simplify any algebraic expression.

Let's look closer at the expression from our problem after applying the distributive property, \(15x - 5 + 12x + 18\). Here, 'like terms' are those that have the same variable part.

• \(15x\) and \(12x\) are like terms (both terms have \(x\))
• \(-5\) and \(18\) are like terms (both are constants)

You add \(15x\) and \(12x\) to get \(27x\). Then, you combine the constants \(-5\) and \(+18\) to get \(+13\).

By combining like terms, the expression becomes truly simplified: \(27x + 13\). This step makes it easier to work with and understand the expression.

Simplification of Expressions

Simplification of algebraic expressions means rewriting an expression in its simplest form without changing its value. It's the culmination of using properties like distribution and combining like terms effectively.

Starting with an expression like \(5(3x - 1) + 6(2x + 3)\), our goal is to perform operations to reduce it to its simplest form. Here's where you bring together all the steps:

• Apply the distributive property to remove parentheses.
• Combine like terms to merge similar parts.

After these steps, the expression gets reduced to \(27x + 13\) with no further simplification possible.

This final form is much more aesthetic and easier to work with in subsequent math problems. Simplification often makes it clearer to see how an expression behaves or to solve an equation when expressions are involved.