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

To solve the linear equation using the transposition method, let's break it down step by step.

### Given Equation:
\[
\frac{3x - 2}{4} - \frac{2}{3} = 2x + 3 - x
\]

### Step 1: Simplify the equation
First, simplify both sides of the equation:

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

This simplifies to:

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

### Step 2: Eliminate fractions by finding a common denominator
To eliminate the fractions, we can multiply each term by the least common multiple (LCM) of the denominators, which in this case is 12.

Multiply every term by 12:

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

This simplifies to:

\[
3(3x - 2) - 8 = 12(x + 3)
\]

### Step 3: Expand and simplify
Now, expand the terms:

\[
9x - 6 - 8 = 12x + 36
\]

Combine like terms:

\[
9x - 14 = 12x + 36
\]

### Step 4: Transpose terms
Now, let's move the terms involving \(x\) to one side and the constant terms to the other side:

\[
9x - 12x = 36 + 14
\]

Simplify:

\[
-3x = 50
\]

### Step 5: Solve for \(x\)
Finally, solve for \(x\) by dividing both sides by \(-3\):

\[
x = \frac{50}{-3} = -\frac{50}{3}
\]

So, the solution to the equation is:

\[
x = -\frac{50}{3}
\]

Would you like further details or have any questions about this solution?

### Related Questions:
1. What is the transposition method, and how does it work?
2. How can you solve the same equation using substitution?
3. How do you handle equations with variables on both sides?
4. What is the significance of clearing fractions in solving equations?
5. How can this method be applied to solve quadratic equations?

### Tip:
When solving linear equations, always aim to clear fractions early on to simplify the equation, making it easier to solve.

Learn how to solve the linear equation 3x - 2/4 - 2/3 = 2x + 3 - x using the transposition method with a step-by-step solution. This problem involves simplifying fractions, combining like terms, and isolating the variable x. Suitable for Grades 9-12.

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