To solve this system of equations by substitution, we are given the two equations:

1. $-4x + 7y = -41$
2. $x = y + 5$

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

We can substitute the expression for $x$ from the second equation into the first equation:

$-4(x) + 7y = -41$ Substitute $x = y + 5$:

$-4(y + 5) + 7y = -41$

Step 2: Simplify the equation

Now expand the terms:

$-4y - 20 + 7y = -41$

Combine like terms:

$3y - 20 = -41$

Step 3: Solve for $y$

Add 20 to both sides of the equation:

$3y = -41 + 20$ $3y = -21$

Now, divide by 3:

$y = \frac{-21}{3} = -7$

Step 4: Substitute $y = -7$ back into the equation $x = y + 5$

Now that we have $y = -7$, substitute this value into the second equation $x = y + 5$:

$x = -7 + 5 = -2$

Final Answer:

The solution to the system is: $x = -2, \quad y = -7$

Would you like any further details or clarification on this solution?

Related Questions:

1. How would you solve the same system by elimination instead of substitution?
2. Can this system have more than one solution?
3. How would the solution change if the second equation were $x = y - 5$?
4. What happens if you substitute $x = 5 - y$ instead of $x = y + 5$?
5. How can you verify the solution by graphing the two equations?

Tip: After solving a system of equations, always substitute your solution back into the original equations to verify it.