Problem 87 - College Algebra

Question

Simplify algebraic expression. \(5(3 x-2)+12 x\)

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression \(5(3 x-2)+12 x\) is \(27x - 10\).

Step 1: Apply Distributive Law

Apply the distributive property to expand the term 5(3 x - 2). This is done by multiplying 5 with each term in the bracket resulting in 15x - 10. So the expression now is \(15x - 10 + 12x\).

Step 2: Collect like terms

After expanding, we get similar terms, **x**, in the equation. Collect these terms together so the equation is now in the form \(15x + 12x - 10\).

Step 3: Simplify

Now, add the coefficients of the like terms. This calculates to 27x - 10, which is the simplified form of the given expression.

Key Concepts

Distributive Property

Understanding the distributive property is essential for simplifying algebraic expressions. Think of distribution as giving out evenly. In algebra, it refers to spreading a number outside the parenthesis across the terms inside. For example, in the expression 5(3x - 2), we apply this property by multiplying the number outside the parenthesis (5) by each term inside the parenthesis (3x and -2).

Here's the process detailed:

• Multiply 5 by 3x to get 15x.
• Multiply 5 by -2 to get -10.

Join these products together, and you get the expanded form 15x - 10. This step transforms the original expression into a simpler form which makes collecting like terms easier!

Collect Like Terms

Once you've used the distributive property, the next step in expression simplification is to collect like terms. Like terms are terms that have the same variable raised to the same power. Essentially, they are 'alike' in their variable parts. In our example, after distribution, we have 15x and 12x as like terms. Combining them is straightforward—you just add or subtract their numerical coefficients depending on their signs.

Here’s what it looks like in action:

• Add the coefficients of 15x and 12x which gives you (15 + 12)x or 27x.
• Keep the -10 as it is since there's no like term to combine it with.

The process results in a more consolidated expression, making it easier to handle and understand.

Algebraic Simplification

After distributing and collecting like terms, the final step is algebraic simplification. This is where you perform the actual addition or subtraction of the coefficients to reach the simplest form of the expression.

For the given problem, once you've collected like terms, you have 27x and -10. Since there are no further like terms to combine, you've already simplified the expression to its simplest form, 27x - 10.

Algebraic simplification helps to not only make expressions shorter and more manageable but also prepares them for evaluation, graphing, or solving equations. Always look out for opportunities to combine like terms and reduce the complexity of algebraic expressions through this method.