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:
- 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.
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Checking (3, -1): \[ 6(3) - 6(-1) = 18 + 6 = 24 \quad (\text{not a solution}) \]
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Checking (-3, 1): \[ 6(-3) - 6(1) = -18 - 6 = -24 \quad (\text{not a solution}) \]
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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:
- Rearranging \( 2x + 9y = -3 \): \[ 9y = -2x - 3 \] \[ y = -\frac{2}{9}x - \frac{1}{3} \]
Now check the provided solutions:
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Checking (0, 3): \[ 2(0) + 9(3) = 0 + 27 = 27 \quad (\text{not a solution}) \]
-
Checking (3, -1): \[ 2(3) + 9(-1) = 6 - 9 = -3 \quad (\text{is a solution}) \]
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Checking (-3, 1): \[ 2(-3) + 9(1) = -6 + 9 = 3 \quad (\text{not a solution}) \]
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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.