To determine which system of equations has the point (5, -3) as a solution, we need to substitute the x-coordinate (5) and the y-coordinate (-3) into each equation and see which one(s) hold true.
Let's evaluate each equation one by one:
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Equation: \(2y = x + 6\)
- Substitute \(x = 5\) and \(y = -3\): \[ 2(-3) = 5 + 6 \implies -6 = 11 \quad \text{(False)} \]
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Equation: \(3y + 6x - 18 = 0\)
- Substitute \(x = 5\) and \(y = -3\): \[ 3(-3) + 6(5) - 18 = 0 \implies -9 + 30 - 18 = 0 \implies 3 = 0 \quad \text{(False)} \]
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Equation: \(y = x - 8\)
- Substitute \(x = 5\): \[ -3 = 5 - 8 \implies -3 = -3 \quad \text{(True)} \]
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Equation: \(2x + y = 7\)
- Substitute \(x = 5\) and \(y = -3\): \[ 2(5) + (-3) = 7 \implies 10 - 3 = 7 \implies 7 = 7 \quad \text{(True)} \]
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Equation: \(y = -x + 6\)
- Substitute \(x = 5\): \[ -3 = -5 + 6 \implies -3 = 1 \quad \text{(False)} \]
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Equation: \(y = 3x - 1\)
- Substitute \(x = 5\): \[ -3 = 3(5) - 1 \implies -3 = 15 - 1 \implies -3 = 14 \quad \text{(False)} \]
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Equation: \(y = x + 2\)
- Substitute \(x = 5\): \[ -3 = 5 + 2 \implies -3 = 7 \quad \text{(False)} \]
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Equation: \(y = x + 5\)
- Substitute \(x = 5\): \[ -3 = 5 + 5 \implies -3 = 10 \quad \text{(False)} \]
The equations that are true with the point (5, -3) are:
- \(y = x - 8\)
- \(2x + y = 7\)
Therefore, the point (5, -3) is a solution to the system of equations consisting of \(y = x - 8\) and \(2x + y = 7\).