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Chapter 2: Problem 43

Solve using the multiplication principle. Don't forget to check! $$ 84=7 n $$

Short Answer

Expert verified
The solution is \(n = 12\).

Step by step solution

01

- Understand the Equation

The given equation to solve is \(84 = 7n\). This equation indicates that 7 times some number \(n\) equals 84.
02

- Apply the Multiplication Principle

To isolate \(n\), divide both sides of the equation by 7. The multiplication principle allows us to perform the same operation on both sides. So, we write: \[\frac{84}{7} = \frac{7n}{7}\]
03

- Simplify Both Sides

Simplify the equation from Step 2. On the left side, \(\frac{84}{7} = 12\). On the right side, \(\frac{7n}{7} = n\). Thus, we have \(12 = n\).
04

- Verify Solution

Substitute \(n = 12\) back into the original equation to check the solution. Calculate \(7 \times 12\) and see if it equals 84. Indeed, \(7 \times 12 = 84\), so \(n = 12\) is correct.

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Key Concepts

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

solving equations
Solving equations in algebra involves finding the value of the variable that makes the equation true. In the given exercise, we need to find the value of \(n\) that satisfies the equation \(84 = 7n\). This equation tells us that 7 times some number \(n\) equals 84. To solve for \(n\), we must isolate it on one side of the equation. By understanding how to manipulate equations, such as using the multiplication principle, we can systematically solve for the unknown variable in a clear and logical manner.
step-by-step solutions
A step-by-step solution helps in breaking down complex problems into manageable parts. For the equation \(84 = 7n\), we follow these steps:
1. Understand the equation: Recognize that 7 times \(n\) equals 84.
2. Apply the multiplication principle: Divide both sides by 7 to start isolating \(n\). This step allows us to simplify the problem systematically:
\[\frac{84}{7} = \frac{7n}{7}\]
3. Simplify both sides: Apply the division to get:
\[\frac{84}{7} = 12\] and \[\frac{7n}{7} = n\]
Thus, we have \(12 = n\). Each step builds upon the previous one, ensuring we solve the problem accurately.
verifying solutions
Verifying solutions is a crucial step in solving equations. It ensures that our answer is correct. In this exercise, we found that \(n = 12\). To verify, we substitute \(n = 12\) back into the original equation \(84 = 7n\):
\[7 \times 12 = 84\]
Calculate the right side and confirm it equals 84:
\[7 \times 12 = 84\]
Therefore, the solution is verified as correct. This final check reassures that our methodology and calculations were accurate. Always ensure to verify your solutions in mathematics to avoid errors.

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