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Chapter 2: Problem 47

Simplify each expression as completely as possible. $$4(x+3 y)+(4 x+3 y)+4 x(3 y)$$

Short Answer

Expert verified
8x + 15y + 12xy.

Step by step solution

01

- Distribute the 4 in the first term

Distribute the 4 to each term inside the parentheses: 4(x + 3y) = 4x + 12y.
02

- Collect like terms

Write out the simplified expression from Step 1 and add the other terms: 4x + 12y + 4x + 3y + 4x(3y).
03

- Add like terms together

Combine the like terms: 4x + 4x = 8x. 12y + 3y = 15y. The expression is now 8x + 15y + 4x(3y).
04

- Multiply in the last term

Multiply 4x by 3y: 4x(3y) = 12xy. The expression is now 8x + 15y + 12xy.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a critical skill in algebra. It involves taking a complex expression and making it as simple as possible. This process often includes performing arithmetic operations and combining like terms.
To simplify means to:
  • Make expressions shorter
  • Reduce the number of terms
  • Make calculations easier
Let's start with the given expression:
\[4(x+3y)+(4x+3y)+4x(3y)\]
We need to follow a series of steps (distributive property, combining like terms, algebraic multiplication) to simplify it fully.
Distributive Property
The distributive property is a key principle that allows you to simplify expressions by distributing a factor across terms inside parentheses. It states:
\(a(b + c) = ab + ac\)
In simpler terms, multiply the factor outside the parentheses by each term within the parentheses.
Using distributive property in our expression:
\[4(x + 3y) = 4x + 12y\]
This means that we multiply 4 by each term inside the parentheses (x and 3y). After distributing the 4, our expression now becomes:
\[4x + 12y + 4x + 3y + 4x(3y)\]
Combining Like Terms
Combining like terms helps to simplify expressions by merging terms that have the same variables raised to the same power.
Like terms have the same variables with identical exponents. For example, 4x and 3x are like terms, but 4x and 4y are not.
Let's identify and combine like terms in our simplified expression:
\[4x + 12y + 4x + 3y + 4x(3y)\]
Combining the like terms, we get:
  • \(4x + 4x = 8x\)
  • \(12y + 3y = 15y\)
Now our expression is:
\[8x + 15y + 4x(3y)\]
Algebraic Multiplication
Algebraic multiplication involves multiplying variables and constants. You often need to multiply both coefficients (the number parts) and variables.
In our expression, we have a term that needs multiplication:
\[4x(3y)\]
We multiply the coefficients 4 and 3 and the variables x and y:
\[4x \times 3y = 12xy\]
After performing this multiplication, our final simplified expression is:
\[8x + 15y + 12xy\]
This final form is simpler and easier to interpret.

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