Problem 64 Determine which numbers in each ... [FREE SOLUTION]

Chapter 1: Problem 64

Determine which numbers in each set are solutions to the corresponding equations. $$ 6(n+2)=23 ;\\{1,3,5\\} $$

Short Answer

Expert verified

No numbers in the set \{1, 3, 5\} are solutions to the equation.

Step by step solution

Substitute Each Number

We have the equation $6(n+2) = 23$ and the set \{1, 3, 5\}. To find which numbers are solutions, we need to substitute each number from the set into the equation and check if it makes the equation true. Let's start with $n = 1$.

Check $n = 1$

Substitute $n = 1$ into the equation: \[ 6(1+2) = 23 \]Simplify the expression inside the brackets: \[ 6 \times 3 = 18 \]Since 18 does not equal 23, $n = 1$ is not a solution.

Check $n = 3$

Substitute $n = 3$ into the equation: \[ 6(3+2) = 23 \]Simplify the expression inside the brackets: \[ 6 \times 5 = 30 \]Since 30 does not equal 23, $n = 3$ is not a solution.

Check $n = 5$

Substitute $n = 5$ into the equation: \[ 6(5+2) = 23 \]Simplify the expression inside the brackets: \[ 6 \times 7 = 42 \]Since 42 does not equal 23, $n = 5$ is not a solution.

Conclusion

None of the numbers from the set \{1, 3, 5\} satisfied the equation $6(n+2) = 23$. Therefore, there are no solutions from this set.

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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 fundamental technique used in solving algebraic equations. It involves replacing a variable with a given number from a set to determine if it satisfies the equation. In our example, we started with the equation $6(n+2) = 23$ and the number set \{1, 3, 5\}.

We substituted each number in the set for $n$ one by one.

For $n = 1$, replacing it in the equation results in $6(1+2)$.
For $n = 3$, it gives $6(3+2)$.
For $n = 5$, the equation becomes $6(5+2)$.

This systematic process helps identify exactly which numbers make the original equation true. It's like testing each possible key with a lock to see if it opens.

The substitution method is very useful for simple equations or to check solutions from a set, as it allows you to confirm the validity of each potential solution.

Checking solutions

Checking solutions is about determining whether a substituted value makes the original equation true. Once you've made a substitution, the next step is to see if the left side of the equation matches the right side. Using our equation $6(n+2) = 23$ and the number set \{1, 3, 5\}, each number was tested:

With $n = 1$, the equation becomes $6 \times 3 = 18$, which does not match 23.
For $n = 3$, it changes to $6 \times 5 = 30$, also not equal to 23.
Finally, with $n = 5$, the multiplication yields $6 \times 7 = 42$, still unequal to 23.

Checking the solutions verifies our equations by carefully comparing both sides, showing us that none of the numbers from the set worked. This precision is crucial in solving mathematical problems accurately and helps avoid errors.

Simplifying algebraic expressions

Simplifying algebraic expressions involves reducing the complexity of an equation by performing operations like addition and multiplication to make calculations easier. This is a critical skill that helps in both solving equations and verifying solutions.

In our example, for each substitution, simplifying actions were performed inside the brackets before multiplying:

For $n = 1$, simplify $(1+2)$ to 3, giving $6 \times 3$.
For $n = 3$, simplify $(3+2)$ to 5, leading to $6 \times 5$.
With $n = 5$, simplify $(5+2)$ to 7, resulting in $6 \times 7$.

These steps transform the original expression into basic arithmetic calculations, making it much easier to see if the resulting number matches the needed value (23 in this case). Properly simplifying expressions is essential for clarity and ease in finding solutions or confirming them.

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91影视

Determine which numbers in each set are solutions to the corresponding equations. $$ 6(n+2)=23 ;\\{1,3,5\\} $$

Short Answer

Step by step solution

Substitute Each Number

Check \(n = 1\)

Check \(n = 3\)

Check \(n = 5\)

Conclusion

Key Concepts

Substitution method

Checking solutions

Simplifying algebraic expressions

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