We are given a set of 8 linear equations in one variable, $x$. We need to solve each equation to find the value of $x$.

1. Problem Description

We are given a set of 8 linear equations in one variable, $x$. We need to solve each equation to find the value of $x$.

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

$4x - 4 = 6 - 2x$

$4x + 2x = 6 + 4$

$6x = 10$

$x = \frac{10}{6} = \frac{5}{3}$

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

$4x - 4 - 2 = 3x$

$4x - 6 = 3x$

$4x - 3x = 6$

$x = 6$

7. $4(1 - 2x) = 3(2 - x)$

$4 - 8x = 6 - 3x$

$-8x + 3x = 6 - 4$

$-5x = 2$

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

8. $3 - 2(2x + 1) = x + 17$

$3 - 4x - 2 = x + 17$

$1 - 4x = x + 17$

$-4x - x = 17 - 1$

$-5x = 16$

$x = -\frac{16}{5}$

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. $x = 6$

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

8. $x = -\frac{16}{5}$