Final answer:
The same (x, y) pair is a solution to both the original and the resulting equation when the equations are linear with proportional coefficients (same slope) and have different constant terms, avoiding parallel lines and ensuring a point of intersection.
Step-by-step explanation:
If you add two equations together, the resulting equation will have the same solution if the system of equations is consistent and independent. When we look at the given equation 6x+7y=32, and we consider adding this to another equation, for the same (x,y) pair to also be a solution, the other equation should be linear and must be not parallel to the given one. This means that two conditions need to be met:
- The equations should have the same slope (i.e., the coefficients in front of the x and y must create the same ratio in both equations).
- The constant term (the numerical term without variables, on one side of the equation) can be different. This difference in constant terms ensures that there is a unique point where the two lines intersect, which corresponds to the solution of the system.
So, the reason the same (x, y) pair is also a solution to the added equation is due to the coefficients being proportional (maintaining the same slope) and the constant term being different, ensuring the lines intersect rather than being parallel.