To solve the system of equations by substitution, we start with the two equations:

1. $y = -7x + 4$
2. $10x - 10y = 40$

Step 1: Substitute $y$ in the second equation.

Substituting $y$ from the first equation into the second equation:

$10x - 10(-7x + 4) = 40$

Step 2: Simplify the equation.

Distributing $-10$:

$10x + 70x - 40 = 40$

Combine like terms:

$80x - 40 = 40$

Step 3: Solve for $x$.

Add $40$ to both sides:

$80x = 80$

Now divide by $80$:

$x = 1$

Step 4: Substitute $x$ back to find $y$.

Now substitute $x = 1$ back into the first equation to find $y$:

$y = -7(1) + 4$

$y = -7 + 4 = -3$

Solution:

The solution to the system is:

$(x, y) = (1, -3)$

Summary

1. $x = 1$
2. $y = -3$

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

1. How would the solution change if we used elimination instead of substitution?
2. What would happen to the solution if the equations represented parallel lines?
3. Can you graph these equations to visually confirm the solution?
4. What are the implications if one of the equations was altered?
5. How can systems of equations be applied in real-life scenarios?

Tip: Always check your solution by plugging the values back into the original equations!