Select all that are true about the system.

Question

systems of equation made up of lines 1 and 2 
line 1 6x-7y=25
line 2 2x+9y=-3
Submit

(
0
,
3
)
(0,3)
is a solution for Line 1 only.

(
3
,
−
1
)
(3,−1)
is a solution to the system.

(
−
6
,
1
)
(−6,1)
is a solution for Line 2 only.

(
−
3
,
1
)
(−3,1)
is a solution to the system.

GPT-4o mini · Answer

To determine which statements are true, we will check each point against both equations of the lines given.

Line 1: \(6x - 7y = 25\)
Line 2: \(2x + 9y = -3\)

We will substitute each point into the equations to see if they satisfy the respective equations.

For the point \((0, 3)\):
- For Line 1:
  \(6(0) - 7(3) = 0 - 21 = -21 \neq 25\)
  (Not a solution for Line 1)
- For Line 2:
  \(2(0) + 9(3) = 0 + 27 = 27 \neq -3\)
  (Not a solution for Line 2)
Conclusion: \((0, 3)\) is not a solution for either line.
For the point \((3, -1)\):
- For Line 1:
  \(6(3) - 7(-1) = 18 + 7 = 25\)
  (Is a solution for Line 1)
- For Line 2:
  \(2(3) + 9(-1) = 6 - 9 = -3\)
  (Is a solution for Line 2)
Conclusion: \((3, -1)\) is a solution to the system (both lines).
For the point \((-6, 1)\):
- For Line 1:
  \(6(-6) - 7(1) = -36 - 7 = -43 \neq 25\)
  (Not a solution for Line 1)
- For Line 2:
  \(2(-6) + 9(1) = -12 + 9 = -3\)
  (Is a solution for Line 2)
Conclusion: \((-6, 1)\) is a solution for Line 2 only.
For the point \((-3, 1)\):
- For Line 1:
  \(6(-3) - 7(1) = -18 - 7 = -25 \neq 25\)
  (Not a solution for Line 1)
- For Line 2:
  \(2(-3) + 9(1) = -6 + 9 = 3 \neq -3\)
  (Not a solution for Line 2)
Conclusion: \((-3, 1)\) is not a solution for either line.

So, the true statements are: