The problem presents a system of linear equations:

$3x + 7y = 10$ $x + y = 2$

You are asked to determine which other system of equations will have the same solution as this one.

Step-by-step solution:

1. Find the solution to the given system: Let's solve the system: $x + y = 2 \quad \text{(Equation 2)}$ Solve for $x$: $x = 2 - y$ Substitute $x = 2 - y$ into the first equation $3x + 7y = 10$: $3(2 - y) + 7y = 10$ Simplify: $6 - 3y + 7y = 10$ $4y = 4$ $y = 1$ Substitute $y = 1$ back into $x = 2 - y$: $x = 2 - 1 = 1$

   So, the solution is $x = 1$ and $y = 1$.

2. Check which of the answer choices have the same solution: We now check each system to see which one also leads to $x = 1$ and $y = 1$:

   • First option: $3x + 7y = 10 \quad \text{and} \quad 2x + 2y = 4$ Substituting $x = 1$ and $y = 1$ into both equations: $3(1) + 7(1) = 10 \quad \text{(True)}$ $2(1) + 2(1) = 4 \quad \text{(True)}$ This system has the same solution.

   • Other options: We can skip checking the other options because the first option already works, but if you'd like, we can verify that no other options match.

Thus, the correct answer is the first option: $\boxed{3x + 7y = 10 \quad \text{and} \quad 2x + 2y = 4}$

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

Here are 5 related questions:

1. What is a general method to solve a system of linear equations?
2. Can a system of linear equations have no solution? When does that happen?
3. How can you determine if two systems of equations have the same solution without solving them?
4. What are the conditions for a system of equations to have infinite solutions?
5. How would the solution change if the second equation was $2x + 3y = 5$?

Tip: Always check if one equation is just a multiple of the other, as this can lead to quick elimination or simplification.