Problem 19 - Algebra for College Students

Question

Solve each equation. \(\frac{n+2}{4}-\frac{2 n-1}{3}=\frac{1}{6}\)

Step-by-Step Solution

Answer

The solution is \(n = \frac{8}{5}\).

Step 1: Find a common denominator

To eliminate the fractions, find the least common multiple (LCM) of the denominators. The denominators are 4, 3, and 6. The LCM of 4, 3, and 6 is 12.

Step 2: Multiply through by the common denominator

Multiply every term in the equation by 12 to clear the fractions.\[12 \left( \frac{n+2}{4} \right) - 12 \left( \frac{2n-1}{3} \right) = 12 \left( \frac{1}{6} \right)\] which simplifies to \[3(n+2) - 4(2n-1) = 2\].

Step 3: Distribute and simplify

Distribute the 3 into \((n+2)\) and the -4 into \((2n-1)\). \[3n + 6 - 8n + 4 = 2\].

Step 4: Combine like terms

Combine the terms involving \(n\) and the constant terms:\[3n - 8n + 6 + 4 = 2\] simplifies to \[-5n + 10 = 2\].

Step 5: Isolate the variable term

Subtract 10 from both sides to get all variable terms on one side and constants on the other:\[-5n = 2 - 10\] which simplifies to \[-5n = -8\].

Step 6: Solve for n

Divide both sides by -5 to solve for \(n\):\[n = \frac{-8}{-5}\] which simplifies to \[n = \frac{8}{5}\].

Key Concepts

Solving Equations

In algebra, solving equations involves finding the value of the unknown variable that makes the equation true. This often means isolating the variable on one side of the equation using a series of steps. Let's use the equation \[ \frac{n+2}{4} - \frac{2n-1}{3} = \frac{1}{6} \] as our example. Here, the goal is to solve for \( n \).

• First, we eliminate fractions by finding a common denominator, which allows us to rewrite the equation without fractions.
• Afterward, we multiply both sides of the equation to clear out these fractions, making it easier to manipulate the terms.
• Then, we distribute any constants across terms in parentheses and simplify by combining like terms.
• Finally, we isolate the variable \( n \) by moving other terms to the opposite side of the equation through addition or subtraction and divide by the coefficient of \( n \) to solve for its value.

Each step methodically brings us closer to pinning down the specific value of the variable, ensuring clarity and accuracy without altering the equation's balance. Solving equations systematically not only hones number-crunching skills but also sharpens logical reasoning.

Fractions in Algebra

Fractions can be intimidating at first glance, but understanding them is key to mastering algebra. When working with fractions in algebraic equations, the crucial task is to manage the fractions so they do not complicate the process:

• Fractions often involve denominators, the numbers below the fractional bar. Simplifying or eliminating them can make handling equations more straightforward.
• To solve equations with fractions, you can find an equivalent expression without the fractions, usually by multiplying by a common denominator.
• This technique helps in avoiding mistakes that might arise from adding or subtracting fractional parts wrongly.

Employing a common denominator simplifies the fractions into whole numbers or integers, which are more convenient to work with in equations. For instance, multiplying each fraction in the equation \[ \frac{n+2}{4} - \frac{2n-1}{3} = \frac{1}{6} \] by the least common multiple of the denominators converts each fraction outright, smoothing the path to the solution. Manipulating fractions efficiently is a vital skill in all areas of algebra and beyond.

Common Denominators

Finding a common denominator is a fundamental step in managing equations involving fractions. By converting fractions to have the same denominator, they become easier to add, subtract, and manipulate.

• The common denominator is typically the least common multiple (LCM) of all the denominators in the equation.
• Once identified, each fraction is adjusted by multiplying both its numerator and denominator, making them equivalent fractions with the common denominator.
• This process is pivotal when dealing with expressions or equations, as it allows you to operate on fractions as though they were standard numbers.

In our given equation \[ \frac{n+2}{4} - \frac{2n-1}{3} = \frac{1}{6} \] , we identified the LCM of the denominators (4, 3, and 6), which is 12. Using 12 as the common denominator simplifies the terms and transitions the equation from an initially complex one into a format that's much easier to resolve. Understanding how to leverage common denominators is an essential toolkit for any aspiring algebra student.