To determine which points are solutions to the system of inequalities \( y > -\frac{2}{5}x + 3 \) and \( y > 4x + 2 \), we will evaluate each point against both inequalities.
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Point \( (12, -7) \):
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For \( y > -\frac{2}{5}x + 3 \): \[ -7 > -\frac{2}{5}(12) + 3 \Rightarrow -7 > -\frac{24}{5} + 3 \Rightarrow -7 > -\frac{24}{5} + \frac{15}{5} \Rightarrow -7 > -\frac{9}{5} \] This is false because \(-7 < -\frac{9}{5}\).
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For \( y > 4x + 2 \): \[ -7 > 4(12) + 2 \Rightarrow -7 > 48 + 2 \Rightarrow -7 > 50 \] This is also false.
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Point \( (2, -3) \):
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For \( y > -\frac{2}{5}x + 3 \): \[ -3 > -\frac{2}{5}(2) + 3 \Rightarrow -3 > -\frac{4}{5} + 3 \Rightarrow -3 > -\frac{4}{5} + \frac{15}{5} \Rightarrow -3 > \frac{11}{5} \] This is false.
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For \( y > 4x + 2 \): \[ -3 > 4(2) + 2 \Rightarrow -3 > 8 + 2 \Rightarrow -3 > 10 \] This is also false.
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Point \( (2, -1) \):
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For \( y > -\frac{2}{5}x + 3 \): \[ -1 > -\frac{2}{5}(2) + 3 \Rightarrow -1 > -\frac{4}{5} + 3 \Rightarrow -1 > -\frac{4}{5} + \frac{15}{5} \Rightarrow -1 > \frac{11}{5} \] This is false.
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For \( y > 4x + 2 \): \[ -1 > 4(2) + 2 \Rightarrow -1 > 8 + 2 \Rightarrow -1 > 10 \] This is also false.
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Point \( (-2, 10) \):
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For \( y > -\frac{2}{5}x + 3 \): \[ 10 > -\frac{2}{5}(-2) + 3 \Rightarrow 10 > \frac{4}{5} + 3 \Rightarrow 10 > \frac{4}{5} + \frac{15}{5} \Rightarrow 10 > \frac{19}{5} \] This is true since \( 10 = \frac{50}{5} > \frac{19}{5} \).
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For \( y > 4x + 2 \): \[ 10 > 4(-2) + 2 \Rightarrow 10 > -8 + 2 \Rightarrow 10 > -6 \] This is also true.
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Therefore, the only point that satisfies both inequalities is \( (-2, 10) \).
The answer is: \( (-2, 10) \)
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