We have the following system of equations:

1. $y - 7.1x = 8.9(x + y) + 8.2$
2. $y + 6.4 = \frac{y - x}{2}$

Step 1: Simplify the second equation.

Start by simplifying the second equation:

$y + 6.4 = \frac{y - x}{2}$

Multiply both sides by 2 to eliminate the fraction:

$2(y + 6.4) = y - x$

This expands to:

$2y + 12.8 = y - x$

Now, subtract $y$ from both sides:

$y + 12.8 = -x$

Rearranging to solve for $x$, we get:

x = -y - 12.8 \tag{3}

Step 2: Substitute equation (3) into the first equation.

Now, substitute $x = -y - 12.8$ into the first equation:

$y - 7.1(-y - 12.8) = 8.9((-y - 12.8) + y) + 8.2$

Step 3: Simplify the first equation.

First, simplify the left side:

$y - 7.1(-y - 12.8) = y + 7.1y + 90.88$

This simplifies further to:

$8.1y + 90.88$

Now, simplify the right side:

$8.9((-y - 12.8) + y) + 8.2 = 8.9(0 - 12.8) + 8.2 = -113.92 + 8.2 = -105.72$

Step 4: Solve for $y$.

Now, set both sides equal:

$8.1y + 90.88 = -105.72$

Subtract 90.88 from both sides:

$8.1y = -105.72 - 90.88$

$8.1y = -196.6$

Solve for $y$:

$y = \frac{-196.6}{8.1} \approx -24.2716$

Step 5: Substitute $y$ back into equation (3) to solve for $x$.

Now, substitute $y = -24.2716$ into equation (3):

$x = -(-24.2716) - 12.8$

$x = 24.2716 - 12.8 = 11.4716$

Final Answer:

The solution for $x$ is approximately $x \approx 11.4716$.

Would you like more details, or do you have any questions?

Here are five related questions:

1. Can you solve for $y$ in greater detail?
2. What if you swap the two equations in the system?
3. How would the solution change if the constants in the equation were different?
4. How do you verify if the solution is correct?
5. Can we solve the system graphically?

Tip: When solving systems of equations, always start by simplifying one equation, then substitute into the other to avoid mistakes!