Problem 36 - 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 Factor

To remove the parentheses, apply the distributive property by multiplying each term inside the parentheses by the factor outside: \(-4\). This yields \(-4 \times 3y + (-4) \times (-4)\).

Step 2: Perform Multiplication

Calculate each of the products from step 1: \(-4 \times 3y = -12y\) and \(-4 \times (-4) = 16\). The expression becomes \(-12y + 16 + 12y\).

Step 3: Combine Like Terms

Combine the like terms in the expression \(-12y + 16 + 12y\). The \(-12y\) and \(+12y\) terms cancel out, simplifying to \(16\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions. It states that multiplying a single term by terms inside a parenthesis involves multiplying the single term by each term within the parentheses individually.
This is usually expressed as:

• \(a(b + c) = ab + ac\)

In our given exercise, we have the expression

• \(-4(3y - 4)\).

Applying the distributive property here means multiplying

• \(-4 \times 3y\) and \(-4 \times (-4)\).

This step is crucial as it breaks the parenthesis, allowing further simplification. It’s often used to rearrange and solve equations or expressions so one can work with them more easily.

Recognizing and Combining Like Terms

Like terms in algebra are terms that include the same variable raised to the same power. They can be combined because they represent the same quantity.
Consider expressions like:

• \(3x\) and \(-7x\),
• \(5y^2\) and \(2y^2\).

In our solution, after applying the distributive property, we arrived at

• \(-12y + 16 + 12y\).

Here,

• \(-12y\) and \(+12y\)

are like terms because they both contain the variable \(y\) raised to the same power (which is 1, although it's usually not written). When combined:

• \(-12y + 12y = 0\),

resulting in no \(y\) terms left in the expression.
This process of combining helps to simplify the expression significantly and makes it easier to solve.

Simplification of Expressions

Simplifying expressions involves reducing them to their most condensed form while maintaining equality. It is an important process in algebra that often involves using the distributive property and combining like terms.
Let's look at the expression from the problem:

• \(-4(3y - 4) + 12y\).

After using the distributive property, we got

• \(-12y + 16 + 12y\).

By recognizing and combining the like terms, the expression simplified further to just

• \(16\).

This final step involves ensuring there are no further operations to be performed, and all terms are expressed in their simplest form. Simplification of expressions is vital in solving equations and making computations more manageable.