To determine which of the provided statements about the system of equations is true, we can evaluate each proposed solution against both equations.
The system of equations is:
- \( 6x - 7y = 25 \) (line 1)
- \( 2x + 9y = -3 \) (line 2)
Checking each option:
a. \( (0, \frac{3}{0}) \)
- Note: The expression \( \frac{3}{0} \) is undefined. So this option is not valid.
b. \( (-6, 1) \)
- Check line 2: \[ 2(-6) + 9(1) = -12 + 9 = -3 \] This is correct, so (-6, 1) is a solution for line 2.
c. \( (3, -1) \)
- Check line 1: \[ 6(3) - 7(-1) = 18 + 7 = 25 \] This holds true.
- Check line 2: \[ 2(3) + 9(-1) = 6 - 9 = -3 \] This holds true as well, so (3, -1) is a solution to the system.
d. \( (-3, 1) \)
- Check line 1: \[ 6(-3) - 7(1) = -18 - 7 = -25 \] This does not equal 25, so it is not a solution for line 1.
Conclusion:
- Option a is invalid due to the undefined value.
- Option b is a valid solution for line 2 only.
- Option c is a valid solution for both lines, hence a solution to the system.
- Option d is not a solution to the system.
The correct answers are b and c, with c being a solution to the entire system.