We are given the following system of equations:

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

To find the value of $x$, we can use substitution since the second equation already expresses $y$ in terms of $x$.

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 to isolate the $x$-term.

$28x = 182$

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

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

So, the value of $x$ is $\boxed{6.5}$.

Would you like to see how to find the corresponding value of $y$, or do you have any other questions?

Related Questions:

1. How do we solve for $y$ in this system?
2. How can you verify the solution for both $x$ and $y$?
3. Can you solve this system using the elimination method?
4. How do you graphically represent the solution of this system of equations?
5. What if the second equation were $y = 5x - 32$, how would the solution change?

Tip: Substitution works best when one of the equations is already solved for one variable, making it easier to substitute and simplify.