Problem 59 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$8(4 y+3)+(-35 y)$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(-3y + 24\).

Step 1: Applying Distributive Property

Expand the expression inside the parenthesis. Distribute the 8 into the parenthesis: \(8*4y + 8*3\). This simplifies to \(32y + 24\). So the equation becomes, \(32y + 24 -35y\)

Step 2: Combining Like Terms

Combine the 'y' terms (32y and -35y) together. This gives: \(32y - 35y + 24\). This simplifies to \(-3y + 24\)

Step 3: Final Simplified Expression

The expression is already simplified. There are no more like terms to combine.

Key Concepts

Distributive Property

The distributive property is a very powerful tool in algebra that lets us simplify expressions and solve equations with ease. When you have an expression like \(8(4y + 3)\), the distributive property allows you to "distribute" the number 8 across each term inside the parentheses.
To apply the distributive property, you multiply the outer number by each term within the parentheses, step-by-step:

• Multiply 8 by the first term, which is \(4y\).
• Multiply 8 by the second term, which is \(3\).

This results in the expression \(32y + 24\). By doing this, you've effectively gotten rid of the parentheses and simplified part of the expression.
This concept is incredibly useful because you can break down more complex expressions into manageable parts, making them easier to work with and understand.

Combining Like Terms

Once we have simplified an expression using the distributive property, the next step is to combine like terms. But what are like terms? Like terms are terms within an expression that have exactly the same variables raised to the same powers. This means that the coefficients (the numerical part) can be added or subtracted.
Let's look at the expression from our previous step: \(32y + 24 - 35y\). To combine like terms:

• Identify terms with the same variable, in this case \(y\).
• Combine the coefficients of these terms.

In our example, \(32y\) and \(-35y\) are like terms. Subtracting \(35y\) from \(32y\) results in \(-3y\). The term \(24\) is not combined with these terms because it doesn't have a \(y\) variable attached.
The process of combining like terms is crucial for simplifying expressions fully so that you can solve equations or evaluate them more easily.

Algebraic Expression

An algebraic expression is essentially a collection of numbers, variables, and operators (like addition and subtraction) put together in a meaningful way. Algebraic expressions represent values or relationships between numbers using symbols like \(x\) or \(y\) to stand for unknown or variable values.
In the expression \(8(4y + 3) + (-35y)\), you are working with a combination of terms and numbers that need to be simplified.
Key components of algebraic expressions include:

• Terms: Parts of the expression separated by plus or minus signs (e.g., \(4y\), \(3\), and \(-35y\)).
• Coefficients: The numerical part of terms with variables (e.g., 8 in \(8(4y + 3)\)).
• Constants: Numbers on their own without variables (e.g., \(3\) or \(24\)).

Simplifying an algebraic expression involves using techniques like the distributive property and combining like terms. This transforms the expression into its simplest form, making it easier to understand and work with.