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Chapter 2: Problem 8

Simplify each expression. $$6(y+2)-4(y-3)$$

Short Answer

Expert verified
The simplified form of the expression is \(2y + 24\).

Step by step solution

01

Apply Distributive Property

By applying the distributive property, we multiply each term inside the parenthesis with the term outside. This gives us: \(6y + 12 - 4y + 12\).
02

Combine Like Terms

Combine 'like' terms together. Here, 'like terms' would mean the terms with the same variable. Therefore, we add together \(6y\) and \(-4y\), and the two constants \(12\) and \(12\), which gives us: \(2y + 24\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a key concept in algebra that allows us to simplify expressions easily. It involves distributing, or multiplying, a single term across terms inside a parenthesis. This is like spreading out a packet of seeds into different pots. In the expression given, \(6(y+2)-4(y-3)\), we distribute 6 across \(y+2\) and -4 across \(y-3\).
  • Start by multiplying 6 by each term inside the first parenthesis: \(6 \times y = 6y\) and \(6 \times 2 = 12\).
  • Then, do the same for -4: \(-4 \times y = -4y\) and \(-4 \times -3 = 12\).
This gives us \(6y + 12 - 4y + 12\). Breaking it down like this makes it easier to handle complex expressions.
It鈥檚 all about distributing one term over others smoothly.
Combining Like Terms
Once you have used the distributive property, the next step in algebra is to simplify by combining like terms. "Like terms" are terms that have the same variable raised to the same power. In our expression \(6y + 12 - 4y + 12\), we have like terms that need to be grouped. Let's break it down:
  • Identify the terms with the variable \(y\): \(6y\) and \(-4y\).
  • Combine them: \(6y - 4y = 2y\).
  • Next, look at the constant terms: 12 and 12.
  • Add them together: \(12 + 12 = 24\).
By doing this, our expression simplifies to \(2y + 24\).
This process can dramatically simplify polynomial expressions and is an essential step in making calculations easier.
Polynomial Expressions
Polynomial expressions are made up of terms, each consisting of a coefficient, a variable, and an exponent. Polynomials can have several terms, such as monomials (one term), binomials (two terms), and trinomials (three terms). The expression \(6(y+2)-4(y-3)\) is a polynomial because it combines multiple terms with variables and constants.
  • Understanding polynomial structure helps in organizing and simplifying problems.
  • Each term in a polynomial expression is influenced by operations like addition, subtraction, multiplication, and division.
  • Knowing the hierarchy of operations and the role of distributive property and like terms is important in tackling polynomial expressions.
Polynomials form the basis for more complex math disciplines.
Strengthening your understanding of them can open the door to mastering other algebraic concepts.

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