Problem 27 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \(x = -12\).

Step 1: Find the least common denominator

The denominators here are 3 and 4. The least common denominator of 3 and 4 is 12. This is because 12 is the smallest number that both 3 and 4 can divide into evenly.

Step 2: Multiply each term by the least common denominator

Multiply each term of the equation by 12 to get rid of the fractions. This results in: \(12*\frac{x}{4} = 12*2 + 12*\frac{x-3}{3}\), which simplifies to \(3x = 24 + 4(x-3)\)

Step 3: Simplify the equation

Simplify the equation by distributing the 4 on the right side of the equation to get \(3x = 24 + 4x - 12\), which simplifies further to \(3x = 4x + 12\).

Step 4: Gather terms with the variable x

Subtract 4x from each side of the equation to get: \(-x = 12\)

Step 5: Solve for x

Multiply each side of the equation by -1 in order to isolate x. This results in \(x = -12\) as the final solution for the equation.

Key Concepts

Understanding the Least Common Denominator

When dealing with fractions, particularly in linear equations, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest number into which all the denominators can evenly divide.
For the equation \(\frac{x}{4}=2+\frac{x-3}{3}\), the denominators are 4 and 3. To solve the equation without fractions, we find their LCD, which is 12. Let's see how:

• The multiples of 4 are: 4, 8, 12, 16,...
• The multiples of 3 are: 3, 6, 9, 12, 15,...
• The smallest common multiple is 12.

This means 12 is the least common denominator, simplifying the process of solving the equation by clearing the fractions.

What Are Denominators and Why They Matter

Denominators are found at the bottom of a fraction and signify how many equal parts make up a whole. In the equation \(\frac{x}{4}=2+\frac{x-3}{3}\), 3 and 4 are the denominators. Adjusting these plays a critical role in solving equations as it allows us to eliminate fractions.
In practice, addressing the denominators involves:

• Identifying all denominator values in the equation.
• Determining their least common denominator (LCD).
• Multiplying the entire equation by the LCD to simplify it by eliminating fractions.

Understanding denominators helps not only in academic exercises but in real-life situations where fractional values are involved.

Solving Linear Equations with Fractions

Linear equations with fractions might look daunting, but they become manageable once you eliminate those fractions. The key is using the least common denominator (LCD).
For the equation \(\frac{x}{4}=2+\frac{x-3}{3}\), we used 12 as the LCD:

• Multiply every term by 12: \(12*\frac{x}{4}=12*2+12*\frac{x-3}{3}\)
• This results in an equation without fractions: \(3x=24+4(x-3)\)
• Simplify the equation until it becomes a standard form: \(3x=4x+12\)
• Isolate the variable x to solve: \(-x=12\)

Finally, multiply by -1 to get \(x=-12\) as the solution. Following these steps ensures that fractions don't stand in the way of solving linear equations.