3x-2 upon 4 -2 upon 3= 2x+3upon 3-x solve the following linear equations using transposition method

Let's solve the given linear equation step by step using the transposition method.

The equation is:

\[
\frac{3x - 2}{4} - \frac{2}{3} = \frac{2x + 3}{3} - x
\]

### Step 1: Eliminate the fractions
To eliminate the fractions, find the least common multiple (LCM) of the denominators, which are 4 and 3. The LCM of 4 and 3 is 12.

Multiply the entire equation by 12 to eliminate the denominators:

\[
12 \times \left(\frac{3x - 2}{4}\right) - 12 \times \left(\frac{2}{3}\right) = 12 \times \left(\frac{2x + 3}{3}\right) - 12 \times x
\]

This simplifies to:

\[
3 \times (3x - 2) - 4 \times 2 = 4 \times (2x + 3) - 12x
\]

### Step 2: Simplify the equation
Now, distribute the constants on both sides:

\[
9x - 6 - 8 = 8x + 12 - 12x
\]

Combine like terms:

\[
9x - 14 = -4x + 12
\]

### Step 3: Transpose the terms
Now, we'll transpose the terms to isolate \(x\) on one side of the equation. First, move the \(-4x\) term to the left side by adding \(4x\) to both sides:

\[
9x + 4x - 14 = 12
\]

This simplifies to:

\[
13x - 14 = 12
\]

Next, move the constant term \(-14\) to the right side by adding 14 to both sides:

\[
13x = 12 + 14
\]

Simplifying further:

\[
13x = 26
\]

### Step 4: Solve for \(x\)
Finally, divide both sides by 13 to solve for \(x\):

\[
x = \frac{26}{13} = 2
\]

### Final Answer:
The solution to the equation is \(x = 2\).

Would you like any additional details or have any questions?

Here are some related questions:
1. How do you handle linear equations with multiple fractions?
2. What is the process for solving linear equations with variables on both sides?
3. Can you explain the importance of transposition in solving equations?
4. How can you check your solution to a linear equation?
5. What is the difference between solving equations using the substitution method versus the transposition method?

**Tip:** When dealing with fractions in linear equations, clearing the denominators by multiplying through by the LCM can simplify the process and reduce errors.

Learn how to solve linear equations involving fractions using the transposition method. This problem covers key concepts in algebra, including eliminating fractions, transposing terms, and solving for variables. Suitable for high school students in Grades 9-12.

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