Which point would be a solution to the system of linear inequalities shown below? \[ y

Ask by Franklin Lane. in the United States

Feb 25,2025

To determine which point is a solution to the system of linear inequalities given by: 1. \( y < x - 7 \) 2. \( y < \frac{1}{5} x - 2 \) we will evaluate the two points provided: \((-10, -2)\) and \((10, 7)\). ### Step 1: Evaluate the point \((-10, -2)\) 1. **For the first inequality \( y < x - 7 \)**: - Substitute \( x = -10 \) and \( y = -2 \): \[ -2 < -10 - 7 \implies -2 < -17 \quad \text{(False)} \] 2. **For the second inequality \( y < \frac{1}{5} x - 2 \)**: - Substitute \( x = -10 \) and \( y = -2 \): \[ -2 < \frac{1}{5}(-10) - 2 \implies -2 < -2 - 2 \implies -2 < -4 \quad \text{(False)} \] ### Conclusion for \((-10, -2)\): Both inequalities are false, so \((-10, -2)\) is **not** a solution. ### Step 2: Evaluate the point \((10, 7)\) 1. **For the first inequality \( y < x - 7 \)**: - Substitute \( x = 10 \) and \( y = 7 \): \[ 7 < 10 - 7 \implies 7 < 3 \quad \text{(False)} \] 2. **For the second inequality \( y < \frac{1}{5} x - 2 \)**: - Substitute \( x = 10 \) and \( y = 7 \): \[ 7 < \frac{1}{5}(10) - 2 \implies 7 < 2 - 2 \implies 7 < 0 \quad \text{(False)} \] ### Conclusion for \((10, 7)\): Both inequalities are false, so \((10, 7)\) is **not** a solution. ### Final Conclusion: Neither point \((-10, -2)\) nor \((10, 7)\) is a solution to the system of linear inequalities.