Problem 7 - Elementary Algebra

Question

Solve each equation. $$5 n-2-8 n=31$$

Step-by-Step Solution

Answer

The solution is n = -11.

Step 1: Combine Like Terms

First, identify and combine the like terms on the left side of the equation. The terms containing 'n' are 5n and -8n. When you combine 5n and -8n, you get \(-3n\). The equation now becomes: \(-3n - 2 = 31\).

Step 2: Isolate the Variable Term

Next, we need to isolate the term containing the variable 'n'. To do this, add 2 to both sides of the equation to eliminate the constant term on the left. This gives us: \(-3n = 31 + 2\), which simplifies to \(-3n = 33\).

Step 3: Solve for the Variable

The last step is to solve for 'n' by dividing both sides of the equation by -3. This gives us: \(n = \frac{33}{-3}\). Simplifying this, we find that \(n = -11\).

Key Concepts

Combining Like Terms

When tackling problems involving linear equations, one of the first steps involves "combining like terms." This means we look for terms that have the same variable raised to the same power. In the equation provided: \[5n - 2 - 8n = 31\]The like terms are the ones involving the variable 'n,' which are \(5n\) and \(-8n\). These terms can be added or subtracted from each other because they share the same variable characteristic.

• This process of combining like terms simplifies your equation, making it more manageable.
• For \(5n - 8n\), calculate \(5 - 8 = -3\).
• The expression thus simplifies to \(-3n\), giving us the new equation: \(-3n - 2 = 31\).

This step is crucial because it reduces the complexity of your equation, helping you to focus only on what is necessary to find the solution.

Isolating Variable Terms

Once like terms are combined, the next step is to work on "isolating variable terms." The aim here is to have only the variable term (in this case \(-3n\)) on one side of the equation, so we can clearly solve for it.In the equation \(-3n - 2 = 31\), we notice that there is a constant \(-2\) on the left side. Our goal is to transfer this constant to the right side to leave \(-3n\) by itself. To achieve

• Add 2 to both sides: \(-3n - 2 + 2 = 31 + 2\).
• This simplifies to \(-3n = 33\).

Isolating the variable terms prepares the equation for the final steps, where you can clearly see what operation needs to be performed to solve for the variable.

Division in Equations

The final step involves "division in equations" to solve for the specific value of the variable. When you're left with an equation like \(-3n = 33\), and you need to find 'n,' the solution involves removing the coefficient of 'n' by dividing.

• The coefficient here is \(-3\), so you divide both sides of the equation by \(-3\).
• This operation fundamentally asks: what is \(33\) divided by \(-3\)?
• By performing the division: \(n = \frac{33}{-3}\), which simplifies to \(n = -11\).

This step is important because it isolates 'n' completely, providing a clear solution. Always check your division to ensure accuracy, as this determines the final answer.