To determine which system of equations has the same solution as the original system:

Original system:

1. $4x + 3y = 19$
2. $2x + 6y = 50$

Step 1: Analyze the equivalent systems

An equivalent system must be derived from the original equations through algebraic manipulations, such as multiplying through by a constant.

Check each option:

1. Option A:
   $-8x + 3y = -38$ and $2x + 6y = 50$
   The first equation does not match the original equation after any consistent manipulation (e.g., scaling or adding). This system is not equivalent.

2. Option B:
   $-8x - 6y = -38$ and $2x + 6y = 50$
   Similarly, scaling or manipulation cannot derive the first equation $-8x - 6y = -38$ from the original system. This is not equivalent.

3. Option C:
   $4x - 6y = -38$ and $2x + 6y = 50$
   If we manipulate $4x + 3y = 19$, subtracting $6y$ from both sides produces $4x - 6y = -38$. Hence, this system is equivalent.

4. Option D:
   $-8x - 6y = 19$ and $2x + 6y = 50$
   Again, $-8x - 6y = 19$ cannot be derived from the original equations. This system is not equivalent.

Correct Answer:

Option C: $4x - 6y = -38$ and $2x + 6y = 50$.

Would you like a detailed explanation or further assistance?

Related Questions:

1. What are the properties of equivalent systems of equations?
2. How do you determine if two systems of equations are consistent?
3. Can matrix transformations be used to check for equivalent systems?
4. Why is subtracting or adding equations a valid operation in systems of equations?
5. How do changes in coefficients affect the solutions of systems?

Tip:

When checking equivalency, always verify by substitution or consistent manipulation to avoid errors.