Solve the following algebraic equations for the unknown variable: 1. 2 − x + 1 = 8x 2. 2x − 8 + 3x = 17 3. (2)/(5)y + 8 = 12 4. Thrice a number divided by 2 results in 10 less than the number.

$$3 = 8x + x$$ $$3 = 9x$$

$$x = \frac{3}{9}$$ $$x = \frac{1}{3}$$

The unknown variable is $\boxed{\frac{1}{3}}$.

2. Find the unknown variable in the following equation: $2x - 8 + 3x = 17$.

$$(2x + 3x) - 8 = 17$$ $$5x - 8 = 17$$

$$5x = 17 + 8$$ $$5x = 25$$

$$x = \frac{25}{5}$$ $$x = 5$$

The unknown variable is $\boxed{5}$.

3. Find the unknown variable in the following equation: $\frac{2}{5}y + 8 = 12$.

$$\frac{2}{5}y = 12 - 8$$ $$\frac{2}{5}y = 4$$

$$y = 4 \times \frac{5}{2}$$ $$y = \frac{20}{2}$$ $$y = 10$$

The unknown variable is $\boxed{10}$.

4. Thrice a number divided by 2 results in 10 less than the number. What is the number?

Translate the problem into an equation.

$$\frac{3n}{2} = n - 10$$

$$3n = 2(n - 10)$$ $$3n = 2n - 20$$

$$3n - 2n = -20$$ $$n = -20$$

The number is $\boxed{-20}$.

5. The product of a number and five is increased by 12, and the result is 32. Find the number.

Translate the problem into an equation.

$$5x + 12 = 32$$

$$5x = 32 - 12$$ $$5x = 20$$

$$x = \frac{20}{5}$$ $$x = 4$$

The number is $\boxed{4}$.