We are asked to solve six linear equations for $x$.

1. Problem Description

We are asked to solve six linear equations for $x$.

2. Solution Steps

1. $x + 3(x+1) = 2x$

$x + 3x + 3 = 2x$

$4x + 3 = 2x$

$4x - 2x = -3$

$2x = -3$

$x = -\frac{3}{2}$

2. $1 + 3(x-1) = 4$

$1 + 3x - 3 = 4$

$3x - 2 = 4$

$3x = 6$

$x = 2$

3. $2x - 2(x+1) = 5x$

$2x - 2x - 2 = 5x$

$-2 = 5x$

$x = -\frac{2}{5}$

4. $2(3x-1) = 3(x-1)$

$6x - 2 = 3x - 3$

$6x - 3x = -3 + 2$

$3x = -1$

$x = -\frac{1}{3}$

5. $4(x-1) = 2(3-x)$

$4x - 4 = 6 - 2x$

$4x + 2x = 6 + 4$

$6x = 10$

$x = \frac{10}{6}$

$x = \frac{5}{3}$

6. $4(x-1) - 2 = 3x$

$4x - 4 - 2 = 3x$

$4x - 6 = 3x$

$4x - 3x = 6$

$x = 6$

3. Final Answer

1. $x = -\frac{3}{2}$

2. $x = 2$

3. $x = -\frac{2}{5}$

4. $x = -\frac{1}{3}$

5. $x = \frac{5}{3}$

6. $x = 6$