Problem 24 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) 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 each term

First, apply the distributive property by multiplying each term inside the parentheses by the factor outside. Do this for both expressions: - For the first expression: \[5(3x - 1) = 5 \cdot 3x + 5 \cdot (-1) = 15x - 5\]- For the second expression: \[6(2x + 3) = 6 \cdot 2x + 6 \cdot 3 = 12x + 18\]

Step 2: Combine expression results

Next, combine the results from the distributions:\[15x - 5 + 12x + 18\]

Step 3: Combine like terms

Combine the terms that are like each other. This involves adding or subtracting coefficients of the same variable or constant:- Combine the \(x\) terms: \[15x + 12x = 27x\]- Combine the constant terms: \[-5 + 18 = 13\]

Step 4: Final Expression

Now, write down the simplified expression from the combined terms:\[27x + 13\]

Key Concepts

Distributive Property

In algebra, one of the key tools we use is the distributive property. It allows us to multiply a single term by two or more terms inside a parentheses. This is incredibly useful when simplifying algebraic expressions.
To understand it better, let's look at the general formula:

• If you have an expression of the form \(a(b+c)\), the distributive property lets you expand it to \(a \cdot b + a \cdot c\).

In our exercise, we apply this property twice. First to the expression \(5(3x-1)\) and then to \(6(2x+3)\). By distributing every term inside the parentheses with the factor outside, we can rewrite complex expressions in simpler, equivalent forms without the parentheses. This step is crucial, as it sets a strong foundation to further simplify algebraic expressions by combining like terms in the next step.

Combining Like Terms

Once terms have been distributed, the next step to simplify the expression involves combining like terms. Like terms are terms that share the same variable raised to the same power; only their coefficients are different.
This step allows us to reduce the expression further by performing mathematical operations on the coefficients of like terms.

• In our exercise, after the distribution, we have the expression: \(15x - 5 + 12x + 18\).
• Combine the terms \(15x\) and \(12x\) because they both have the variable \(x\), resulting in \(27x\).
• Similarly, combine the constant terms \(-5\) and \(18\), which yields \(13\).

Combining like terms helps to make expressions more concise and easier to interpret, ensuring you've gathered all similar variables and constants together.

Simplifying Expressions

Simplifying expressions is the end goal of using both the distributive property and combining like terms. It allows for a clearer, more concise way to represent algebraic equations. After you have expanded and combined all like terms, you arrive at a simplified expression.
The steps we've covered lead us to

• a streamlined expression that is easier to evaluate or compare with other expressions.

In our example, following the steps of distribution and combining like terms, the final simplified expression is \(27x + 13\). This expression is free of parentheses and has collected all like variables and constants, making it as simple as possible to work with in further calculations or algebraic manipulations. Simplifying expressions not only aids in solving equations but also enhances understanding of mathematical relationships within algebra.