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Chapter 2: Problem 52

Solve each equation. Check each solution. $$ -3 n-\frac{1}{3}=\frac{8}{3} $$

Short Answer

Expert verified
The solution to the equation is \(n = -1\).

Step by step solution

01

Isolate the variable term

To solve the equation \(-3n - \frac{1}{3} = \frac{8}{3}\), the first step is to isolate the term containing \(n\). We can do this by adding \(\frac{1}{3}\) to both sides of the equation:\[-3n - \frac{1}{3} + \frac{1}{3} = \frac{8}{3} + \frac{1}{3}\]This simplifies to:\[-3n = \frac{9}{3}\]which further simplifies to:\[-3n = 3\].
02

Solve for the variable

Now that we have \(-3n = 3\), we need to solve for \(n\). We do this by dividing both sides by \(-3\):\[n = \frac{3}{-3}\]Simplifying the fraction, we get:\[n = -1\].
03

Check the solution

To ensure \(n = -1\) is a solution, substitute \(n\) back into the original equation \(-3n - \frac{1}{3} = \frac{8}{3}\).Substitute \(n = -1\):\[-3(-1) - \frac{1}{3} = \frac{8}{3}\]Simplifying the left side gives:\[3 - \frac{1}{3} = \frac{8}{3}\]Convert 3 to a fraction with a denominator of 3:\[\frac{9}{3} - \frac{1}{3} = \frac{8}{3}\]This simplifies to:\[\frac{8}{3} = \frac{8}{3}\]Since both sides are equal, \(n = -1\) is indeed the correct solution.

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Key Concepts

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

Isolating the Variable
One of the most important steps when solving equations is to isolate the variable. This means getting the variable on one side of the equation by itself.
In the equation \(-3n - \frac{1}{3} = \frac{8}{3}\),
we initially have \(-3n\) combined with a fraction.
To isolate \(-3n\),
we need to move the \(-\frac{1}{3}\) to the other side.Here's how you can do it:
  • Add \(\frac{1}{3}\) to both sides of the equation to cancel out the \(-\frac{1}{3}\).
    This results in:\[-3n - \frac{1}{3} + \frac{1}{3} = \frac{8}{3} + \frac{1}{3}\]
  • After simplification, you'll have:\[-3n = \frac{9}{3}\]
  • This simplifies further to:\(-3n = 3\)
This step is crucial because it makes it easier to see what operations are necessary to solve for the variable next.
Solving for the Variable
Once you've isolated the variable term, the next step is to solve for the variable. In this example, you want to solve for \(n\) in \(-3n = 3\).The best approach here is:
  • Divide both sides of the equation by \(-3\) to solve for \(n\).
    This results in:\[n = \frac{3}{-3}\]
  • This simplifies to:\[n = -1\]
By dividing both sides by the coefficient of \(n\),
you isolate \(n\) completely.
This is the core of solving for any variable in an equation.
Checking Solutions
After you've found a solution, it's important to check your work. This ensures that the solution satisfies the original equation and is not just a numerical slip. For the equation \(-3n - \frac{1}{3} = \frac{8}{3}\),
we found that \(n = -1\).
Let's verify:Substitute \(n = -1\) back into the original equation:
  • \[-3(-1) - \frac{1}{3} = \frac{8}{3}\]
  • Simplifying, you get:\[3 - \frac{1}{3} = \frac{8}{3}\]
  • Convert 3 to a fraction:\[\frac{9}{3} - \frac{1}{3} = \frac{8}{3}\]
  • These simplify to:\[\frac{8}{3} = \frac{8}{3}\]
Since both sides equal, \(n = -1\) is indeed correct.
Always substitute to verify; it ensures accuracy and reinforces understanding of equations.

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