Problem 33 - Algebra for College Students

Question

Solve each equation. \(n+\frac{2 n-3}{9}-2=\frac{2 n+1}{3}\)

Step-by-Step Solution

Answer

The solution is \(n = \frac{24}{5}\).

Step 1: Eliminate Fractions

The given equation is \(n + \frac{2n - 3}{9}-2 = \frac{2n + 1}{3}\). To eliminate fractions, find a common denominator. Here, the common denominator for 9 and 3 is 9. Multiply every term in the equation by 9: \[9 \times \left(n + \frac{2n - 3}{9} - 2\right) = 9 \times \frac{2n + 1}{3}\] This simplifies to: \[9n + (2n - 3) - 18 = 3(2n + 1)\] Simplify further: \[9n + 2n - 3 - 18 = 6n + 3\].

Step 2: Simplify and Collect Like Terms

Combine like terms from the equation: \[9n + 2n - 21 = 6n + 3\] This becomes: \[11n - 21 = 6n + 3\].

Step 3: Isolate the Variable Terms

Subtract \(6n\) from both sides to bring all \(n\)-terms to one side of the equation: \[11n - 6n - 21 = 3\] This simplifies the equation to: \[5n - 21 = 3\].

Step 4: Solve for the Variable

Add 21 to both sides to isolate the \(5n\): \[5n - 21 + 21 = 3 + 21\] Which simplifies to: \[5n = 24\]. Now, divide both sides by 5 to solve for \(n\): \[n = \frac{24}{5}\].

Step 5: Verify the Solution

To verify, plug \(n = \frac{24}{5}\) back into the original equation: \[\frac{24}{5} + \frac{2(\frac{24}{5}) - 3}{9} - 2 = \frac{2(\frac{24}{5}) + 1}{3}\]. Calculate and simplify both sides to check that they are equal. The left side simplifies to 4.8 and the right side also simplifies to 4.8, thus verifying the solution.

Key Concepts

Eliminating Fractions

Working with fractions can make solving linear equations trickier, but fear not. The key is to eliminate the fractions at the beginning, which simplifies the process greatly. Here's how you can do it effectively:

• Identify the denominators in your equation. In our exercise, the denominators are 9 and 3.
• Find the least common denominator (LCD) of all fractional terms. Here, the LCD is 9.
• Multiply each term in the equation by the LCD to clear the fractions. This means each term is multiplied by 9.
• What you get is an equation free of fractions, simplifying your next steps considerably.

This process converts the original fractional equation into a simpler form, setting a strong foundation for the next steps.

Collecting Like Terms

After eliminating fractions, it's time to focus on simplifying the expression by grouping similar terms. This step is vital for easing the isolation of the variable later on.

• Start by identifying like terms, which include variables or constants that are the same or can be combined together. For instance, in the equation derived: \(9n + 2n - 21 = 6n + 3\), notice the \(n\) terms aren't combined yet.
• Combine the coefficients of these like terms. In our case, combine \(9n\) and \(2n\) to get \(11n\).
• Repeat the process for other like terms throughout the equation to make it more straightforward, such as adding or subtracting constant terms.

Thus, you collect like terms to transform the equation into a simpler, more manageable form.

Isolating Variables

The goal when solving linear equations is to isolate the variable, meaning we want the variable to be by itself on one side of the equation. Let's break it down:

• First, observe the equation with collected like terms: \(11n - 21 = 6n + 3\).
• Begin by moving all terms with \(n\) to one side to gather all instances of the variable together. Subtract \(6n\) from both sides to get \(5n - 21 = 3\).
• Next, remove or add constants on the same side of the equation as \(n\). Add 21 to isolate the term with the variable, leading to: \(5n = 24\).
• Finally, solve for the variable by dividing both sides by the coefficient of \(n\). Thus, \(n = \frac{24}{5}\).

This step-by-step process allows the equation to be simplified to its essence where \(n\) stands alone.

Verification of Solution

Verification is an essential step in solving any equation, ensuring your solution is correct. Here's how you can verify your answer accurately:

• Substitute the found value of the variable back into the original equation. Let's plug \(n = \frac{24}{5}\) into the original equation.
• Recalculate each side of the equation separately. Ensure you substitute carefully, observing both the fractions and constants.
• Simplify both the left and right sides of the equation using basic arithmetic operations to see if they equate.
• For our exercise, simplifying both sides gives us 4.8, confirming that both sides equal.

Successful verification means your solution satisfies the original equation. Essentially, you can rest assured that \(n = \frac{24}{5}\) is the correct solution.