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Chapter 5: Problem 81

Solve each equation. Check your solution. $$14=8+3 n$$

Short Answer

Expert verified
The solution is \(n = 2\).

Step by step solution

01

Isolate the Variable

Begin by isolating the term that contains the variable. In the equation \(14 = 8 + 3n\), we want to isolate \(3n\). To do this, subtract 8 from both sides of the equation: \(14 - 8 = 3n\). This simplifies to \(6 = 3n\).
02

Solve for the Variable

Now, solve for \(n\) by dividing both sides of the equation by 3: \(\frac{6}{3} = \frac{3n}{3}\). This simplifies to \(2 = n\).
03

Check the Solution

To verify the solution, substitute \(n = 2\) back into the original equation: \(14 = 8 + 3(2)\). Simplify the right-hand side: \(8 + 6 = 14\). Since both sides are equal, the solution \(n = 2\) is correct.

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Key Concepts

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

Isolating the Variable
Understanding the importance of isolating the variable is crucial to solving linear equations. In the equation \(14 = 8 + 3n\), our goal is to have the variable term \(3n\) alone on one side of the equation. Moving terms around to achieve this is called isolating the variable.

Here's how it works:
  • Identify the term containing the variable. In our case, it is \(3n\).
  • Move any other numbers in the same side to the opposite side of the equation. We do this by performing the inverse operation. Since 8 is added to \(3n\), you subtract 8 from both sides.
  • The equation simplifies: Subtracting 8 gives \(14 - 8 = 3n\), or \(6 = 3n\).
This leaves you with \(3n\) isolated, making it easier to find the value of \(n\). Isolating the variable sets the stage for solving the equation directly.
Checking the Solution
Checking your solution is an essential step each time you solve an equation. It ensures your answer is correct, giving you confidence in your solution.

Here's how you can check the solution:
  • Take the value you found for the variable, in this case \(n = 2\), and substitute it back into the original equation \(14 = 8 + 3n\).
  • Replace the variable with your solution: \(8 + 3(2)\).
  • Simplify the right side: \(8 + 6\) simplifies to \(14\).
  • Verify: Since both sides of the equation \(14 = 14\) are equal, your solution is verified.
Checking ensures accuracy and is especially crucial when solving more complex equations. It's like double-checking your work to be extra certain of its correctness.
Step by Step Solution
Approaching a math problem with a step by step solution helps break down complicated processes into manageable chunks. It's similar to following a recipe; each step leads you to the final 'dish' or answer.

The step-by-step approach:
  • First, focus on simplifying the equation by isolating variables (as explained earlier).
  • Next, solve the simplified equation to determine the value of the variable. For \(6 = 3n\), divide both sides by 3 to find \(n\): \(\frac{6}{3} = \frac{3n}{3}\), which simplifies to \(2 = n\).
  • Finally, always check your solution, as detailed before.
By following these structured steps, you not only solve the problem but also develop a clearer understanding of the mechanics of equation solving. Each step builds on the last, leading you to a successful resolution.

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