Problem 64 - College Algebra

Question

simplify each algebraic expression. $$ 4(2 y-6)+3(5 y+10) $$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(23y + 6\).

Step 1: Application of Distributive Property on Each Parenthesis

Distribute 4 into the terms in the first parenthesis and 3 into the terms in the second parenthesis. This gives the new expression: \(8y - 24 + 15y + 30\)

Step 2: Combine Like Terms

Look through the expression and add or subtract the like terms. In this instance, our like terms are the constants (numbers) and the variables ('y' terms). Combining gives us the expression: \(8y + 15y - 24 + 30\)

Step 3: Simplify Further

Calculating the sum of the 'y' terms as well as the constants will give the simplified expression. Therefore, the final simplified expression is: \(23y + 6\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. This property is very useful because it simplifies expressions and equations making them easier to handle. In mathematical terms, the distributive property states that for any numbers or expressions \(a\), \(b\), and \(c\),

• \(a(b + c) = ab + ac\)
• \(a(b - c) = ab - ac\)

Let's look at an example: if you have the expression \(4(2y - 6)\), you apply the distributive property by multiplying 4 with both \(2y\) and \(-6\). This results in the expression \(8y - 24\). Similarly, for \(3(5y + 10)\), you distribute 3 to both \(5y\) and \(10\), obtaining \(15y + 30\). By performing these multiplications, the expression changes from something with parentheses into a series of simpler terms that we can work with further.

Combining Like Terms

Combining like terms is a crucial skill when dealing with algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power. This helps to simplify the expression, making it easier to solve or evaluate.
For our specific expression \(8y - 24 + 15y + 30\), the like terms are the terms that contain 'y', which are \(8y\) and \(15y\), and the constant terms \(-24\) and \(30\). You can only combine terms that are alike, so you'll group them by their type:

• Combine the 'y' terms: \(8y + 15y = 23y\)
• Combine the constants: \(-24 + 30 = 6\)

Therefore, combining these like terms results in the expression \(23y + 6\). A helpful hint is to always look for and group similar variable terms and constants together for easier calculations.

Simplification in Algebra

Simplification in algebra refers to the process of transforming an expression into its most reduced form. Through simplification, you end up with a more compact expression that is easier to read and solve. The main goal is reducing complexity while maintaining the equivalence of the expression.
In the example we explored, after applying the distributive property and combining like terms, we got the expression \(23y + 6\). This is the simplest form of the original expression \(4(2y - 6) + 3(5y + 10)\).
During simplification, it's important to do each step methodically:

• Always start with distributing to remove any grouping symbols like parentheses.
• Next, identify and combine like terms.
• Finally, perform any easy additions or subtractions to simplify the expression further.

The key to mastering simplification is practice and being able to recognize when an expression is fully simplified. Once you grasp these steps, solving algebraic problems becomes much more straightforward.