Problem 58 - Elementary Algebra

Question

\(\frac{n+1}{2}+\frac{n}{3}=\frac{1}{2}\)

Step-by-Step Solution

Answer

The value of \( n \) is 0.

Step 1: Eliminate Fractions

To eliminate the fractions, first find the least common denominator (LCD) of the fractions involved: \(2\), \(3\), and the numbers \(\frac{1}{2}\). The LCD is 6. Multiply all terms by 6 to eliminate the fractions: \[ 6 \left( \frac{n+1}{2} \right) + 6 \left( \frac{n}{3} \right) = 6 \left( \frac{1}{2} \right) \]This simplifies to:\[ 3(n+1) + 2n = 3 \]

Step 2: Distribute and Simplify

Now, distribute the numbers through the parentheses and combine like terms:\[ 3n + 3 + 2n = 3 \]Combine the \(n\) terms:\[ 5n + 3 = 3 \]

Step 3: Isolate the Variable

Subtract 3 from both sides to isolate the \(n\) term:\[ 5n = 0 \]

Step 4: Solve for n

Divide both sides by 5 to solve for \(n\):\[ n = 0 \]

Key Concepts

Eliminating Fractions

Fractions in equations can make them look complex, but one simple technique can help: eliminating them altogether. When we encounter fractions in a linear equation, it means that the terms of the equation have different denominators. The trick here is to multiply every part of the equation by a common value that removes the fractions.

This value is known as the "least common denominator" (LCD), which we'll discuss in the next section. Multiplying through by the LCD will clear the equation of fractions, allowing us to work with whole numbers or integers. This step significantly simplifies our calculation process and makes the algebraic manipulation of the terms easier.

Least Common Denominator

When dealing with fractions in an equation, finding the Least Common Denominator (LCD) is a crucial step. The LCD is the smallest number that each denominator can divide without leaving a remainder. It effectively levels the playing field by clearing the fractions.

Consider the equation:

• The denominators are 2, 3, and 2 (from \( \frac{n+1}{2} \), \( \frac{n}{3} \), and \( \frac{1}{2} \)).
• The LCD for these numbers is 6.

By multiplying each term of the equation with 6, we eliminate the fractions, leading us to a simpler equation to solve. This approach is commonly used in solving equations involving fractions and is fundamental to maintaining equation equality.

Combining Like Terms

Once fractions are eliminated, combining like terms is necessary to simplify the equation further and bring similar elements together. In algebra, like terms are terms that have the same variable raised to the same power. By consolidating them, we reduce the complexity of expressions.

For instance, after eliminating fractions in our example, the equation

• Changes from: \(3(n + 1) + 2n = 3\)
• To: \(3n + 3 + 2n = 3\)
• We then combine the \(n\) terms (\(3n\) and \(2n\)) to obtain: \(5n + 3 = 3\).

These simplifications make the equation more manageable and pave the way for isolating the variable in the next step. Remember, the goal is to make the equation as simple as possible while preserving its equality.

Isolating Variable

Once we've simplified our equation by combining like terms, the next step is to isolate the variable (in this case, \(n\)). Isolating the variable means getting the variable by itself on one side of the equation.

This allows us to clearly see the solution. For our example:

• We have the equation: \(5n + 3 = 3\).
• Subtract 3 from both sides to clear any constants from the side with \(n\): \(5n = 0\).
• Finally, divide each side by 5 to solve for \(n\): \(n = 0\).

This isolation of the variable lets us find the solution easily. In this example, the solution to the equation is \(n = 0\). Isolating the variable is an essential part of solving linear equations and can be achieved using basic operations like addition, subtraction, multiplication, and division while keeping the equation balanced.