Let's solve the two linear equations shown in the image one by one:

1. First equation:

$\frac{z}{5} = \frac{3}{10} z + 8$

Step 1: Eliminate the fractions. Multiply both sides by 10 to eliminate the denominators: $10 \times \frac{z}{5} = 10 \times \left( \frac{3}{10} z + 8 \right)$ This simplifies to: $2z = 3z + 80$

Step 2: Solve for $z$. Subtract $3z$ from both sides: $2z - 3z = 80$ This gives: $-z = 80$

Now, divide both sides by $-1$: $z = -80$

So, the solution to the first equation is: $z = -80$

2. Second equation:

$2(1 - x) = 7(1 + 2x) + 2$

Step 1: Expand both sides. Expand the left and right sides: $2 - 2x = 7 + 14x + 2$ Simplify the right-hand side: $2 - 2x = 9 + 14x$

Step 2: Collect like terms. Move all terms involving $x$ to one side and constants to the other: $2 - 9 = 14x + 2x$ Simplify: $-7 = 16x$

Step 3: Solve for $x$. Divide both sides by 16: $x = \frac{-7}{16}$

So, the solution to the second equation is: $x = \frac{-7}{16}$

Would you like further details on any step or a more thorough explanation? Here are some related questions:

1. What method can you use to eliminate fractions in linear equations?
2. How do you handle variables on both sides of an equation?
3. Why is it important to simplify both sides of the equation before solving?
4. What are some common mistakes when distributing terms in equations?
5. Can a linear equation ever have no solution or infinitely many solutions?

Tip: Always double-check your work by substituting your solution back into the original equation to ensure it's correct!