Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation and check your proposed solution. Begin your work by rewriting each equation without fractions. \(\frac{3 x}{4}-3=\frac{x}{2}+2\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 20\).

Step 1: Identify the Common Denominator

The denominators here are 4 and 2. The least common denominator is 4.

Step 2: Multiply Each Term by the Common Denominator

Multiplying each term by 4 to remove the fractions, the equation becomes \(3x - 12 = 2x + 8\).

Step 3: Solve for x

By collecting like terms, it becomes \(x = 20\).

Step 4: Check the Solution

Substitute x = 20 back into the original equation. If both sides of the equation are equal, then the answer is correct. \(\frac{3 \cdot 20}{4}-3 =\frac{20}{2}+2 \) simplifies to \(15 - 3 = 10 + 2\) and finally to \(12 = 12\). Since the equation holds true, the solution is correct.

Key Concepts

Understanding the Least Common Denominator

When dealing with equations that involve fractions, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest number that can be evenly divided by all the denominators in the equation. This step is essential to eliminate fractions, making the equation easier to work with.

• Identify the denominators in the equation. In our example, they are 4 and 2.
• Determine the least common denominator by finding the smallest multiple shared by these numbers. Here, the LCD is 4.

By multiplying each term by the LCD, you ensure that the fractions are eliminated, simplifying the equation into a more manageable form for further steps. Understanding and effectively finding the LCD is a key skill in solving equations with fractions.

Eliminating Fractions in Equations

Once the least common denominator is determined, the next step is eliminating fractions by multiplying every term in the equation by this number. This process clears the fractions, thereby simplifying the equation.Start by applying the LCD to each term:

• Multiply each term by the LCD (which is 4 in our example) to eliminate denominators.
• The equation then transforms: from \(\frac{3x}{4} - 3 = \frac{x}{2} + 2\) to \(3x - 12 = 2x + 8\).

By doing so, each fraction disappears, leaving an equation that consists solely of integers. This makes it much easier to solve as there are no fractions involved, allowing you to focus on finding the value of the variable by isolating it.

Checking Solutions for Accuracy

To verify the accuracy of your solution, checking the result by substituting it back into the original equation is a vital step. This confirmation step ensures that the solution is correct and that no errors were made during the calculation.Follow these guidelines to check your solution:

• Substitute the proposed value of the variable back into the original equation.
• Calculate both sides of the equation separately.
• Ensure both sides yield the same result when simplified.

Using our example, substituting \(x = 20\) into \(\frac{3x}{4} - 3 = \frac{x}{2} + 2\), both sides reduce to 12, showing that the solution holds.Always checking your solutions helps avoid mistakes, confirming that each step was executed correctly.