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Chapter 2: Problem 103

Simplify by combining like terms whenever possible. $$3(x+y)+4(x-y)$$

Short Answer

Expert verified
7x - y

Step by step solution

01

Distribute the Constants

First, distribute the constants 3 and 4 to both terms inside the parentheses. It will look like this: 3(x+y) + 4(x-y) = 3x + 3y + 4x - 4y
02

Combine Like Terms

Combine the like terms by adding or subtracting the coefficients of similar variables (x terms and y terms). It will look like this: 3x + 4x + 3y - 4y = (3x + 4x) + (3y - 4y)
03

Final Simplification

Simplify the expression by performing the arithmetic operations on the coefficients: 3x + 4x = 7x 3y - 4y = -y So, the simplified expression is 7x - y.

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Key Concepts

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

Combining Like Terms
In algebra, you often need to simplify expressions by combining like terms. Like terms are terms that have the same variables raised to the same power. For example, in the expression \(3x + 4x\), both terms are like terms because they both contain the variable \(x\) raised to the same power.
To combine like terms, you add or subtract the coefficients (the numbers in front of the variables). This is a key step in making expressions easier to work with and is often used in solving equations.
Consider the expression \(3x + 4x + 3y - 4y\). Here's how combining like terms works:
  • Group the terms with the same variable: \((3x + 4x) + (3y - 4y)\).
  • Add or subtract the coefficients of these terms: \(3x + 4x = 7x\) and \(3y - 4y = -y\).

After combining like terms, the simplified expression becomes \(7x - y\).
Distribution in Algebra
The distributive property is a fundamental algebraic property used to multiply a single term by two or more terms inside parentheses. It ensures that you multiply the term outside the parentheses by each term within the parentheses.
Consider the exercise \(3(x + y) + 4(x - y)\). Here's how you use distribution:
  • Multiply \(3\) by both \(x\) and \(y\): \(3(x + y) = 3x + 3y\).
  • Multiply \(4\) by both \(x\) and \(y\), noting the negative sign with \(y\): \(4(x - y) = 4x - 4y\).

After distribution, the expression becomes \(3x + 3y + 4x - 4y\). This step is crucial before you can combine like terms and further simplify the expression.
Coefficients in Algebra
Coefficients are the numerical part of terms in an algebraic expression. They tell you how many times to count the variable. For instance, in the term \(3x\), the coefficient is \(3\), meaning 3 times \(x\).
Here's why understanding coefficients is important:
  • They help you in combining like terms by adding or subtracting the numerical parts.
  • They play a key role in distribution, where you multiply the coefficients by each term inside the parentheses.

In our initial expression \(3(x + y) + 4(x - y)\), the coefficients \(3\) and \(4\) were distributed to the terms inside the parentheses. Finally, combining the like terms \(3x\) and \(4x\) resulted in \(7x\), while \(3y\) and \(-4y\) resulted in \(-y\).
Understanding coefficients allows you to simplify and solve algebraic expressions accurately and efficiently.

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