Question

Line 1: 6x-6y=25
Line2: 2x+9y=-3
(0,3) is the only solution for line 1 only
(3,-1) is a solution to the system (-3,1) is a solution to the system
(-6,1) is a solution for line 2 only

Answer 1

To analyze the information provided about the two lines, let's first rewrite the equations in a more manageable form and check the given solutions.

The equations of the two lines are:

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

### Step 1: Verify the Solutions for Line 1

We can rewrite Line 1 into slope-intercept form to easily analyze solutions:

1. Rearranging \( 6x - 6y = 25 \):
\[
-6y = -6x + 25
\]
\[
y = x - \frac{25}{6}
\]

Now verify the solutions provided:

- **(0, 3)**:
\[
6(0) - 6(3) = 0 - 18 = -18 \quad (\text{not a solution})
\]

Since (0, 3) is stated to be a solution, it seems incorrect based on the calculation.

- Checking **(3, -1)**:
\[
6(3) - 6(-1) = 18 + 6 = 24 \quad (\text{not a solution})
\]

- Checking **(-3, 1)**:
\[
6(-3) - 6(1) = -18 - 6 = -24 \quad (\text{not a solution})
\]

- Checking **(-6, 1)**:
\[
6(-6) - 6(1) = -36 - 6 = -42 \quad (\text{not a solution})
\]

### Step 2: Check Solutions for Line 2

Next, rewrite Line 2 into slope-intercept form:

1. Rearranging \( 2x + 9y = -3 \):
\[
9y = -2x - 3
\]
\[
y = -\frac{2}{9}x - \frac{1}{3}
\]

Now check the provided solutions:

- Checking **(0, 3)**:
\[
2(0) + 9(3) = 0 + 27 = 27 \quad (\text{not a solution})
\]

- Checking **(3, -1)**:
\[
2(3) + 9(-1) = 6 - 9 = -3 \quad (\text{is a solution})
\]

- Checking **(-3, 1)**:
\[
2(-3) + 9(1) = -6 + 9 = 3 \quad (\text{not a solution})
\]

- Checking **(-6, 1)**:
\[
2(-6) + 9(1) = -12 + 9 = -3 \quad (\text{is a solution})
\]

### Summary of Solutions
- Solutions analyzed do verify:
- **(3, -1)**: is a solution to **Line 2**.
- **(-6, 1)**: is a solution to **Line 2** only.
- **(0, 3)**: is not a solution for either line.
- **(-3, 1)**: is not a solution for either line.

Based on the tests above:
- Line 1 has the correct form and no confirmed unique solutions as described initially.
- If more accurate results for both equations are necessary, double-check original equations or asserted solutions.