Problem 26 - College Algebra Essentials

Question

Contain linear equations with constants in denominators. Solve equation. \(\frac{x+1}{4}=\frac{1}{6}+\frac{2-x}{3}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\)

Step 1: Identify the least common multiple (LCM)

Firstly, identify the LCM of the denominators 4, 6, and 3. Here, the LCM is 12.

Step 2: Multiply each term with the LCM

Multiply both sides of the equation by the LCM to eliminate the denominators. It will result in \(12*(\frac{x+1}{4})=12*(\frac{1}{6}+\frac{2-x}{3})\) that simplifies to \(3(x+1)=2+4*(2-x)\).

Step 3: Simplify the equation

Now simplify the equation at both sides. The equation after simplifying will be \(3x+3=2+8-4x\). Further, simplify it to \(3x+3=10-4x\).

Step 4: Regroup like terms

Bring the terms involving 'x' to one side of the equation and constant terms to the other side of the equation. This leads to \(3x+4x=10-3\), which simplifies to \(7x=7\).

Step 5: Solve for x

Finally, solve for 'x' by dividing by 7 on both sides of the equation. It gives the solution as \(x=1\).

Key Concepts

Least Common Multiple (LCM)

When working with linear equations that have fractions, finding the least common multiple (LCM) is a crucial step. The LCM of a set of numbers is the smallest multiple that they all share. For the denominators in the exercise \(\frac{x+1}{4}=\frac{1}{6}+\frac{2-x}{3}\), we need to find the LCM of 4, 6, and 3.

Here’s how you can find the LCM:

• List the multiples of each number until you find the smallest common one. For 4, 6, and 3, the LCM is 12.
• This step is vital because multiplying each term by the LCM will eliminate the fractions and simplify the equation into a form that's easier to solve.

Denominators

Denominators are the bottom parts of fractions. In the context of equations, denominators tell us how to divide numbers evenly. For example, in our exercise, 4, 6, and 3 are the denominators.

Understanding denominators is key:

• They indicate into how many parts a whole is divided.
• When solving an equation, the goal is often to "clear" these denominators to simplify the process.

This is accomplished by multiplying all terms in the equation by the LCM of these denominators, transforming the equation into a linear form without fractions.

Solving Equations Step-by-Step

Solving equations step-by-step is like peeling back the layers of an onion, revealing the solution at the core. Let's break down the process for our equation:

1. **Multiply by the LCM:** Begin by multiplying each term by the LCM (12), which clears the fractions and simplifies the equation.
2. **Simplify:** After eliminating fractions, simplify by expanding and combining like terms on both sides.
3. **Regroup Terms:** Move terms containing the variable \(x\) to one side and constants to the other to form an equation like \(7x=7\).
4. **Solve for \(x\):** Finally, solve for \(x\) by dividing both sides by the coefficient of \(x\).

This systematic approach ensures clarity and structure, making it easier to arrive at an accurate solution.