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Chapter 1: Problem 75

Solve each equation. $$ \frac{3}{4} x-5=\frac{2}{3} x+\frac{1}{4} $$

Short Answer

Expert verified
The solution to the equation is \(x = 63\).

Step by step solution

01

Eliminate Fractions

To get rid of the fractions, find the least common multiple (LCM) of the denominators 4 and 3. The LCM is 12. Multiply every term by 12 to eliminate the fractions:\[ 12 \times \left( \frac{3}{4}x - 5 \right) = 12 \times \left( \frac{2}{3}x + \frac{1}{4} \right) \] Simplifying gives:\[ 9x - 60 = 8x + 3 \]
02

Isolate Variable "x"

Subtract \(8x\) from both sides of the equation to bring the terms containing \(x\) to one side:\[ 9x - 8x - 60 = 3 \]This simplifies to:\[ x - 60 = 3 \]
03

Solve for "x"

Add 60 to both sides to solve for \(x\):\[ x - 60 + 60 = 3 + 60 \]This simplifies to:\[ x = 63 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Least Common Multiple
To solve an equation with fractions, it might be beneficial to clear them out completely. Fractions often make things look complicated. One effective way to get rid of fractions is by finding the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of each of the denominators. For example, when dealing with fractions that have denominators 4 and 3, the LCM is 12. Here's how it works:

  • Find the smallest number that 4 (like in the fraction \( \frac{3}{4} \)) and 3 (like in the fraction \( \frac{2}{3} \)) both divide into without leaving a remainder.
  • This number is 12 because 12 can be divided evenly by both 4 and 3.
Once you have the LCM, you multiply all terms in the equation by this number to remove the fractions. This simplifies the equation, making it much easier to handle.
Eliminating Fractions from the Equation
After determining the least common multiple, you multiply each term of the equation by this number. This step helps in eliminating fractions from the equation:

When you multiply \( \frac{3}{4}x - 5 \) and \( \frac{2}{3}x + \frac{1}{4} \) by 12, you distribute the 12 across all the parts of the equation:

  • 12 \cdot \frac{3}{4}x = 9x
  • 12 \cdot -5 = -60
  • 12 \cdot \frac{2}{3}x = 8x
  • 12 \cdot \frac{1}{4} = 3
So, the equation \( \frac{3}{4} x - 5 = \frac{2}{3} x + \frac{1}{4} \) becomes \( 9x - 60 = 8x + 3 \). This process clears the fractions and simplifies the equation a bunch, setting you up for the next step.
Isolating Variables to Solve
The next step after eliminating fractions is to isolate the variable. This means getting all terms with the variable on one side of the equation and constants on the other. In our equation \( 9x - 60 = 8x + 3 \), follow these simple steps:

  • First, subtract \( 8x \) from both sides to bring all the \( x \)-related terms to one side: \( 9x - 8x - 60 = 3 \).
  • This simplifies to \( x - 60 = 3 \).
  • Now, to solve for \( x \), you need to undo the subtraction of 60 by adding 60 to both sides: \( x - 60 + 60 = 3 + 60 \).
  • This finally simplifies to \( x = 63 \).
By performing these operations, you have successfully isolated \( x \) and solved the equation! Remember, the key is to perform same operations on both sides to keep the equation balanced.

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