To determine which points are solutions to the given lines, we need to substitute the coordinates of each point into the equations of both lines and see if they satisfy the equations.
Line 1: \( 6x - 7y = 25 \)
Line 2: \( 2x + 9y = -3 \)
Now let's test each point:
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Point (0, 3) for Line 1: \[ 6(0) - 7(3) = 0 - 21 = -21 \quad (\text{not a solution}) \] This point is not a solution for Line 1.
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Point (-3, 1) for both lines: For Line 1: \[ 6(-3) - 7(1) = -18 - 7 = -25 \quad (\text{not a solution}) \] For Line 2: \[ 2(-3) + 9(1) = -6 + 9 = 3 \quad (\text{not a solution}) \]
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Point (-6, 1) for Line 2: For Line 2: \[ 2(-6) + 9(1) = -12 + 9 = -3 \quad (\text{is a solution}) \] For Line 1: \[ 6(-6) - 7(1) = -36 - 7 = -43 \quad (\text{not a solution}) \]
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Point (3, -1) for both lines: For Line 1: \[ 6(3) - 7(-1) = 18 + 7 = 25 \quad (\text{is a solution}) \] For Line 2: \[ 2(3) + 9(-1) = 6 - 9 = -3 \quad (\text{is a solution}) \]
Summary:
- (0, 3): Not a solution for Line 1
- (-3, 1): Not a solution for either line
- (-6, 1): Solution for Line 2 only
- (3, -1): Solution for the system of both lines
Therefore, the valid conclusions are:
- Point (0, 3) is a solution only for Line 1: False
- Point (-3, 1) is a solution to the system: False
- Point (-6, 1) is a solution for Line 2 only: True
- Point (3, -1) is a solution to the system: True
3 answers