We are given the following system of equations:

1. $3x - 5y = 22$
2. $y = -5x + 32$

To solve for $x$, we'll substitute the expression for $y$ from the second equation into the first equation.

Step 1: Substitute $y = -5x + 32$ into the first equation:

$3x - 5(-5x + 32) = 22$

Step 2: Distribute the -5 across the terms inside the parentheses:

$3x + 25x - 160 = 22$

Step 3: Combine like terms:

$28x - 160 = 22$

Step 4: Add 160 to both sides:

$28x = 182$

Step 5: Solve for $x$ by dividing both sides by 28:

$x = \frac{182}{28} = 6.5$

Thus, the value of $x$ is $6.5$.

The correct answer is: 6.5

Would you like further details or explanations?

Here are five related questions to deepen your understanding:

1. How can you check if your solution is correct by substituting back into both equations?
2. What is the value of $y$ corresponding to $x = 6.5$?
3. What is substitution method in solving systems of equations?
4. What other methods can be used to solve systems of equations, besides substitution?
5. How can you tell if a system of equations has no solution or infinitely many solutions?

Tip: Always double-check your solutions by plugging the values of the variables back into the original equations.