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Chapter 7: Problem 17

Simplify. (Assume all denominators are nonzero.) $$ 23 y-46-1 y $$

Short Answer

Expert verified
The expression simplifies to \( 22y - 46 \).

Step by step solution

01

Identify Like Terms

Begin by identifying like terms in the expression \( 23y - 46 - 1y \). Like terms are terms that contain the same variable raised to the same power. Here, \( 23y \) and \( -1y \) are like terms because they both contain the variable \( y \).
02

Combine Like Terms

Combine the like terms \( 23y \) and \( -1y \) by adding their coefficients. The coefficients are 23 and -1, so the combination is \( (23 - 1)y = 22y \).
03

Simplify the Expression

With the like terms combined, rewrite the expression as \( 22y - 46 \). This is the simplified form of the original expression.

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Key Concepts

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

Like Terms
When working with algebraic expressions, recognizing like terms is crucial. Like terms are terms that have identical variables raised to the same powers. In the expression \( 23y - 46 - 1y \), the terms \( 23y \) and \( -1y \) are considered like terms because they both contain the variable \( y \). The term \(-46\) is not a like term with the others because it does not contain the variable \( y \).

Identifying like terms is the first step in simplifying expressions effectively.
  • You can only combine terms that have the same variable components.
  • The power to which the variable is raised significantly affects the identification of like terms.
Recognizing like terms allows us to simplify expressions and solve algebraic equations more easily.
Coefficients
Coefficients are the numerical factors attached to variables within algebraic expressions. They tell us how many times the variable is being multiplied. In our example, the coefficients are 23 and -1, corresponding to the terms \( 23y \) and \(-1y\) respectively.

Understanding coefficients is key in algebraic manipulations:
  • The coefficient 23 means that \( y \) is expressed 23 times, while \(-1\) signifies that \( y \) is subtracted once.
  • When combining terms, only their coefficients are summed or subtracted while the variable part remains unchanged.
This understanding helps you deal with more complex expressions by focusing on the numbers and the variables separately before merging them.
Combining Like Terms
Combining like terms is the process of simplifying an expression by summing or subtracting the coefficients of these terms. For the expression we have, \( 23y - 1y \) turns into \((23 - 1)y = 22y\) through this process, simplifying the expression.

Here's how you can combine like terms effectively:
  • Group all like terms first, which allows you to see what needs to be combined.
  • Add or subtract their coefficients, leaving the variable part untouched.
  • Ensure you manage the signs correctly; a common mistake is neglecting a negative sign.
After combining like terms, any extraneous constants are left in the expression, which leads to a simplified result, as seen here with the final expression becoming \( 22y - 46 \). This reduced form is much simpler, making it easier to work with in further calculations or equations.

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