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Chapter 8: Problem 267

In the following exercises, solve the equation. $$ n-12=32 $$

Short Answer

Expert verified
n = 44

Step by step solution

01

Understand the Problem

The given equation is \(n - 12 = 32\). The goal is to solve for \(n\).
02

Isolate the Variable

To isolate \(n\), add 12 to both sides of the equation: \(n - 12 + 12 = 32 + 12\).
03

Simplify Both Sides

Simplify the equation after adding 12 to both sides: \(n = 44\).
04

Verify the Solution

To ensure the solution is correct, substitute \(n = 44\) back into the original equation: \(44 - 12 = 32\), which is true.

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Key Concepts

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

Isolate the Variable
When solving linear equations, the first crucial step is to isolate the variable. This means getting the variable (in this case, it鈥檚 'n') on one side of the equation by itself. In the equation given \(n - 12 = 32\), the variable 'n' is currently on the left side but it has '-12' subtracted from it. To isolate 'n', you need to undo this subtraction by performing the opposite operation, which is addition.
  • Step 1: Identify the term that needs to be moved to the other side (here, it's '-12').
  • Step 2: Perform the opposite operation to both sides of the equation (add 12 to both sides).
By doing so, you will have:
\[n - 12 + 12 = 32 + 12\]This simplifies to:
\[n = 44\]Now, you have successfully isolated the variable.
Simplify the Equation
Once you have isolated the variable, it鈥檚 time to simplify the equation. Simplifying means performing any arithmetic operations to get the final value of the variable.
In our example, after adding 12 to both sides, the equation becomes:
\[n = 44\]
Here, there are no further operations left to perform, and 'n' is already by itself. Thus, the equation is simplified to the final solution:
\[n = 44\]
Remember, simplifying an equation often involves combining like terms, reducing fractions, or performing other arithmetic operations to make the equation as straightforward as possible.
Verify the Solution
The final step in solving linear equations is to verify the solution. This means checking your work to ensure the solution you found is correct. To verify, substitute the variable back into the original equation and see if both sides are equal.
For our example, substitute \(n = 44\) back into the original equation:
\[44 - 12 = 32\]
Both sides of the equation are equal, confirming that \(n = 44\) is indeed the correct solution.
  • Double-check your arithmetic to make sure there were no mistakes in your calculations.
  • If both sides of the equation are equal, your solution is correct.
Verification is a crucial step because it ensures accuracy and boosts your confidence in your mathematical skills.

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