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Chapter 2: Problem 1

Simplify each expression by combining any like terms. See Examples 3 and 4. $$ 7 y+8 y $$

Short Answer

Expert verified
\(7y + 8y = 15y\).

Step by step solution

01

Identify Like Terms

Look at the expression and identify the terms that have the same variable raised to the same power. In this case, both terms, \(7y\) and \(8y\), have the variable \(y\), making them like terms.
02

Add the Coefficients

Combine the like terms by adding their coefficients. Coefficients are the numbers in front of the variables. For \(7y + 8y\), the coefficients are 7 and 8. Add these together: \(7 + 8 = 15\).
03

Write the Simplified Expression

Attach the variable \(y\) to the result of the addition of the coefficients. Thus, \(7y + 8y\) simplifies to \(15y\).

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Key Concepts

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

Understanding Like Terms
In algebra, the process of simplifying expressions often involves dealing with 'like terms.' Like terms are terms that contain the same variable raised to the same power. They are like peas in a pod鈥攖hey look similar because they share the same variable components.
For instance, in the expression \(7y + 8y\), both terms have the variable \(y\). Even if the coefficients (the numbers multiplying the variable) are different, the terms themselves are 'like' because of the variable they share.
  • Terms with only numbers (constants) are considered like terms with other constants.
  • Terms like \(3x^2\) and \(5x^2\) are like terms because both include the variable \(x\) raised to the power of 2.
Recognizing which terms are like terms is crucial. This is the first step in the simplification process of any variable expression.
Combining Coefficients
Once like terms are identified, the next step in simplification is 'combining coefficients.' Coefficients are the numbers placed in front of variables. They tell us how many of the variable we are dealing with.
For example, in the expression \(7y + 8y\), the terms \(7y\) and \(8y\) have coefficients 7 and 8, respectively. These coefficients are combined by simply adding them together: \[7 + 8 = 15\]
This process is akin to counting items: if you have 7 apples and acquire 8 more apples, the total becomes 15 apples. The same goes for variables; you just add up the coefficients while maintaining the variable part.
Working with Variable Expressions
Variable expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. When simplifying these expressions, you often work with like terms and combine coefficients as discussed above.
Variable expressions like \(7y + 8y\) may initially seem complex. Breaking them down into their components helps demystify them.
  • Understand that variables symbolize unknown values and are denoted by letters like \(x\), \(y\), etc.
  • Realize that every step in simplifying is logical and follows set rules that ensure you're solving correctly.
  • Practice recognizing patterns鈥攆or instance, identifying like terms, to make the entire process smoother.
With practice, simplifying variable expressions becomes as straightforward as basic arithmetic.

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