Final answer:
The system of equations -5x 4y=-7 and 17x-16y=31 has one solution: x = 59/37 and y = 31/37.
Step-by-step explanation:
To determine if the system of equations -5x + 4y = -7 and 17x - 16y = 31 has no solutions, infinitely many solutions, or one solution, we can solve the equations using the method of elimination.
Multiplying the first equation by 4 and the second equation by -1, we eliminate the y variable:
-20x + 16y = -28
-17x + 16y = -31
Adding the two equations together, we get:
-37x = -59
Dividing by -37, we find that x = 59/37.
Substituting this value of x back into one of the original equations, we can solve for y:
-5(59/37) + 4y = -7
After simplifying, we find that y = 31/37.
Therefore, the system of equations has one solution: x = 59/37 and y = 31/37.