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Chapter 1: Problem 61

Determine which numbers in each set are solutions to the corresponding equations. $$ 6 n+2=26 ;\\{0,2,4\\} $$

Short Answer

Expert verified
The solution is \( n = 4 \).

Step by step solution

01

Substitute Each Number

Let's test each number from the set \( \{0,2,4\} \) one by one in the equation \( 6n + 2 = 26 \). We start by substituting \( n = 0 \).
02

Check Solution for n=0

Substitute \( n = 0 \) into the equation: \[ 6(0) + 2 = 0 + 2 = 2. \] Since \( 2 eq 26 \), \( n = 0 \) is not a solution.
03

Check Solution for n=2

Substitute \( n = 2 \) into the equation: \[ 6(2) + 2 = 12 + 2 = 14. \] Since \( 14 eq 26 \), \( n = 2 \) is not a solution.
04

Check Solution for n=4

Substitute \( n = 4 \) into the equation: \[ 6(4) + 2 = 24 + 2 = 26. \] Since \( 26 = 26 \), \( n = 4 \) is a solution.
05

Conclusion

After testing all the numbers in the set \( \{0,2,4\} \), the only number that satisfies the equation \( 6n + 2 = 26 \) is \( n = 4 \).

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Key Concepts

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

Substitution Method
The substitution method is a valuable technique used to solve equations, especially linear equations. It involves plugging each value from a given set into the equation to check if it satisfies the equation. This method is straightforward and allows us to verify possible solutions quickly. The basic steps are:
  • Select a number from a set of potential solutions.
  • Substitute this number into the equation wherever the variable appears.
  • Calculate to determine if the substitution makes the equation true (i.e., both sides equal).
  • If true, the number is a solution; if false, try the next number.
This process is efficient for checking solutions and is especially common in mathematics when evaluating specific numbers within a set. It's excellent for students because it provides a clear way to verify their understanding and see where they might have made a mistake.
Linear Equations
Linear equations are mathematical statements that involve variables raised only to the first power, like the equation in the exercise: \(6n + 2 = 26\). These equations represent a straight line when graphed on a coordinate plane, hence the name 'linear.' Working with linear equations has a few key concepts:
  • Each term that includes a variable has a coefficient (e.g., in \(6n\), 6 is the coefficient).
  • They can have one or more variables, but each variable will appear singly.
  • Solutions to these equations are the values of the variables that make the equation true (e.g., finding \(n = 4\) as a solution).
Linear equations are fundamental in algebra and they provide a foundation for understanding more complex mathematical concepts. Learning to solve them accurately is an essential skill.
Mathematical Solutions
Finding solutions to mathematical problems involves determining the values that satisfy a given equation. In our exercise, we are looking for the correct value of \(n\) that makes the equation \(6n + 2 = 26\) true. To identify the right solution:
  • We start with potential solutions, in this case, the set \(\{0, 2, 4\}\).
  • We use methods like substitution to test each candidate from the set.
  • The ultimate goal is to find which candidate, if any, results in a true statement when substituted back into the equation.
  • In our example, only \(n = 4\) satisfies the equation, making it the mathematical solution.
Solving equations helps build analytical skills and fosters a deeper understanding of mathematical relationships and problem-solving strategies.

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Most popular questions from this chapter

The following table shows a few of the airports in the United States with the largest volumes of passengers. Complete this table. The first line is completed for you. (Source: Airports Council International) $$ \begin{array}{|l|c|c|c|c|} \hline & \text { Total passengers in 2015 } & \text { Amount Written } & \text { Standard Form } & \text { Standard Form } \\ & \text { (in hundred-thousands } & \text { in Standard } & \text { Rounded to the } & \text { Rounded to the } \\ \text { City Location of Airport } & \text { of passengers) } & \text { Form } & \text { Nearest Million } & \text { Nearest Ten-Million } \\ \hline \text { Atlanta, GA } & 1015 & 101,500,000 & 102,000,000 & 100,000,000 \\\ \hline \text { Los Angeles, CA } & 749 & & & \\ \hline \end{array} $$

In Example 12 in this section, we found that the average of \(75,96,81,\) and 88 is \(85 .\) If the number 96 is removed from the list of numbers, does the average increase or decrease? Explain why.

Write each whole number in expanded form. $$66,049$$

$$ \begin{array}{r} 19 \\ 214 \\ 49 \\ +651 \\ \hline 923 \end{array} $$

Write each whole number in expanded form. $$47,703,029$$
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