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Chapter 9: Problem 28

Substitute \(1,2,3,\) and 4 for \(n\) in each of the following expressions. $$3 n+1$$

Short Answer

Expert verified
The results are 4, 7, 10, and 13 for \(n\) values of 1, 2, 3, and 4, respectively.

Step by step solution

01

Substitute 1 for n

Replace the variable \(n\) with 1 in the expression \(3n+1\). This gives us: \[3(1) + 1 = 3 + 1 = 4\]
02

Substitute 2 for n

Replace \(n\) with 2 in the expression \(3n+1\). This results in: \[3(2) + 1 = 6 + 1 = 7\]
03

Substitute 3 for n

Substitute \(n\) with 3 in the expression \(3n+1\). This becomes: \[3(3) + 1 = 9 + 1 = 10\]
04

Substitute 4 for n

Replace \(n\) with 4 in the expression \(3n+1\). This gives us: \[3(4) + 1 = 12 + 1 = 13\]

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Key Concepts

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

Evaluating Expressions
Evaluating expressions is a fundamental concept in algebra that helps to find the value of an expression by substituting a number for a variable. When evaluating an expression like \(3n+1\), you replace the variable \(n\) with a specific number. Let鈥檚 consider how this process works step by step:
  • First, choose a number to substitute for the variable. For example, in the expression \(3n+1\), if you choose \(n=2\), substitute 2 wherever you see \(n\).
  • The expression now becomes \(3(2)+1\), which simplifies to \(6+1\). You then add these numbers together to get 7.
Evaluating expressions helps to verify the behavior of algebraic equations with different numbers substituted in. It plays a crucial role in finding specific outcomes based on different inputs.
Algebraic Manipulation
Algebraic manipulation involves altering and rearranging algebraic expressions to make them more manageable. In the context of substituting numbers into expressions, you perform algebraic manipulation by:
  • Substituting numbers into an expression, such as \(3n+1\).
  • Doing arithmetic operations like multiplication and addition to simplify the expression.
  • Continuing these operations until you reach a simplified numerical result.
For instance, let鈥檚 take \(3n+1\) again. If \(n=3\), substituting gives \(3(3)+1\). You multiply: \(3 \times 3 = 9\), and then add 1 to make it 10. Such manipulations are not only helpful in solving tasks like this but are also widely used to solve equations and inequalities.
Prealgebra Concepts
Understanding prealgebra concepts lays the groundwork for more complex algebraic operations. Beginning with simple tasks like substitution, students start seeing patterns and relationships between numbers and symbols. Here are some fundamental aspects:
  • Substitution: This is the act of replacing a variable with a number. It鈥檚 an early step in prealgebra that helps learners get comfortable working with variables.
  • Arithmetic Operations: Performing basic calculations such as multiplication and addition in expressions helps in understanding the order and importance of operations.
  • Constants and Variables: Recognizing the difference between constants (like 1 in \(3n+1\)) and variables (like \(n\)) is crucial. This distinction forms the basis of crafting and solving equations.
Mastering these prealgebra concepts ensures a firm foundation, enabling students to delve deeper into algebra with confidence. They also gain skills necessary for tackling more challenging topics in mathematics later on.

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Most popular questions from this chapter

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(4 \times 10^{-5}\right)\left(9 \times 10^{-10}\right)}{6 \times 10^{-6}}$$

Simplify the following expressions. \(\left(9 a^{5} b^{6}\right)^{2}\left(8 a^{5} b\right)\)

Multiply the following expressions. $$(7 x-3)(7 x+3)$$

Find each quotient. Write all answers in scientific notation. $$\frac{0.00092}{46,000}$$

Expand and multiply. $$(5 x-2)^{2}$$
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