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Chapter 3: Problem 8

Solve each equation. $$5 n-6=19$$

Short Answer

Expert verified
The solution is \(n = 5\).

Step by step solution

01

Understand the Equation

The given equation is \(5n - 6 = 19\). Our goal is to solve for \(n\). We need to isolate \(n\) on one side of the equation.
02

Add to Both Sides

To isolate the term with \(n\), start by eliminating the constant term on the left side. Add 6 to both sides of the equation: \(5n - 6 + 6 = 19 + 6\). This simplifies to \(5n = 25\).
03

Divide Both Sides

Now, divide both sides of the equation by 5 to solve for \(n\): \(\frac{5n}{5} = \frac{25}{5}\). This simplifies to \(n = 5\).

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Key Concepts

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

Isolating Variables
When solving linear equations like \(5n - 6 = 19\), the primary goal is to isolate the variable, which is represented by \(n\) in this equation. Isolating a variable means getting it alone on one side of the equation. This requires you to undo the operations being done to the variable. Remember, you want to treat the equation like a balanced scale: whatever you do to one side, you must do to the other as well.
  • Identify the variable you need to solve for. Here, \(n\) is our target variable.
  • Look at the operations affecting this variable. In this equation, the operations are multiplication by 5 and subtraction by 6.
Keep these points in mind as you proceed to reverse these operations one by one to get \(n\) on its own.
Addition Property of Equality
The addition property of equality is a fundamental rule used to keep equations balanced. It states that you can add the same number to both sides of an equation without changing its equality. In our example, the equation \(5n - 6 = 19\) needs modification to isolate \(n\).
To do this, we start by removing \(-6\), which is subtracted from the \(5n\). By adding \(6\) to both sides of the equation, we effectively eliminate the \(-6\) beside \(n\):
  • Original equation: \(5n - 6 = 19\)
  • Add 6 to both sides: \(5n - 6 + 6 = 19 + 6\)
  • Simplify both sides: \(5n = 25\)
This step helps to gradually bring \(n\) to a solitary position on its side of the equation.
Division Property of Equality
After using the addition property of equality, we are left with a simpler equation: \(5n = 25\). The next task is to directly solve for \(n\). Here, the division property of equality comes into play. This property lets you divide both sides of an equation by the same non-zero number without affecting the equality.
Since \(n\) is multiplied by 5 in \(5n = 25\), we'll divide both sides by 5 to isolate \(n\):
  • Original simplified equation: \(5n = 25\)
  • Divide both sides by 5: \(\frac{5n}{5} = \frac{25}{5}\)
  • Result: \(n = 5\)
By applying this property, you've effectively solved for \(n\), demonstrating the power of balancing equations through division.

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