Problem 35 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ -4(3 y-4)+12 y $$

Step-by-Step Solution

Answer

The simplified expression is 16.

Step 1: Distribute the -4

To remove the parentheses, distribute the -4 to each term inside the parentheses. Multiply -4 by 3y and -4 by -4. This gives you: \[ -4 imes 3y + (-4) imes (-4) = -12y + 16 \]

Step 2: Combine Like Terms

Now that the parentheses are removed, combine the like terms in the expression. You have \(-12y\) from the distributed part and \(12y\) from the original expression. Combine \(-12y\) and \(12y\): \[ -12y + 12y + 16 = 0y + 16 = 16 \]

Step 3: Simplify the Expression

At this point, the only terms remaining are zero and a constant. The expression simplifies to just a constant: \[ 16 \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us to remove parentheses and simplify expressions. When you see an expression like \[-4(3y-4)\], it means you need to distribute, or multiply, -4 by each term inside the parentheses.

Here's how you do it:

• Multiply -4 by 3y. This gives us -12y.
• Next, multiply -4 by -4. This yields positive 16 because multiplying two negative numbers results in a positive number.

Putting it all together, the expression -4(3y-4) simplifies to -12y + 16. Distributing is like spreading a number evenly across terms. It sets the stage for the next step, which is combining those terms together.

Combining Like Terms

Once you have distributed all terms, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. For example, \(-12y\) and \(12y\) are like terms because they both contain the variable y.

In the expression -12y + 12y + 16, you can combine the \(y\) terms:

• Add -12y and 12y together. These cancel each other out because they are opposites, resulting in 0y.
• The constant 16 remains unchanged as there are no other constant terms to combine it with.

Combining like terms narrows down the expression by eliminating unnecessary parts, leaving a cleaner, more manageable expression. In this case, it leaves us with just 16.

Simplifying Expressions

Simplifying expressions is like cleaning up a room; it makes your mathematical work more organized and easier to understand. After using the distributive property and combining like terms, you want to ensure your result is in the simplest form possible. In our example, you end with:
-12y + 12y + 16, which reduced to 0y + 16. Simplifying removes the 0y since zero multiplied by anything is zero.

By simplifying, you verify that you have no unnecessary terms. Your final expression is a neat, simple 16. Remember that simplifying is the process of ensuring your result has the fewest terms and straightforward constants or coefficients.

• Eliminate any terms that equal zero.
• Make sure there are no parentheses remaining.
• Check that all like terms are combined.

By following these steps, you ensure that your algebraic expression is as tidy and simple as it can be.