Which of the following points shows a solution to the simultaneous inequalities y>12x+5

To find which of the points shows a solution to the simultaneous inequalities \(y > 12x + 5\), \(y > \frac{1}{2}x + 5\), and \(y < -2x + 1\), we will test each point in the inequalities.

1. **Test the point \((4, 8)\)**:
- \(y > 12x + 5\): \(8 > 12(4) + 5\) → \(8 > 53\) (False)
- \(y > \frac{1}{2}x + 5\): \(8 > \frac{1}{2}(4) + 5\) → \(8 > 2 + 5\) → \(8 > 7\) (True)
- \(y < -2x + 1\): \(8 < -2(4) + 1\) → \(8 < -8 + 1\) → \(8 < -7\) (False)

This point does not satisfy all three inequalities.

2. **Test the point \((-1.6, 4.2)\)**:
- \(y > 12x + 5\): \(4.2 > 12(-1.6) + 5\) → \(4.2 > -19.2 + 5\) → \(4.2 > -14.2\) (True)
- \(y > \frac{1}{2}x + 5\): \(4.2 > \frac{1}{2}(-1.6) + 5\) → \(4.2 > -0.8 + 5\) → \(4.2 > 4.2\) (False; but it's not strictly greater)
- \(y < -2x + 1\): \(4.2 < -2(-1.6) + 1\) → \(4.2 < 3.2 + 1\) → \(4.2 < 4.2\) (False; but it's not strictly less)

This point does not satisfy all three inequalities.

3. **Test the point \((-5, 5)\)**:
- \(y > 12x + 5\): \(5 > 12(-5) + 5\) → \(5 > -60 + 5\) → \(5 > -55\) (True)
- \(y > \frac{1}{2}x + 5\): \(5 > \frac{1}{2}(-5) + 5\) → \(5 > -2.5 + 5\) → \(5 > 2.5\) (True)
- \(y < -2x + 1\): \(5 < -2(-5) + 1\) → \(5 < 10 + 1\) → \(5 < 11\) (True)

This point satisfies all three inequalities.

4. **Test the point \((1, 5.5)\)**:
- \(y > 12x + 5\): \(5.5 > 12(1) + 5\) → \(5.5 > 12 + 5\) → \(5.5 > 17\) (False)
- \(y > \frac{1}{2}x + 5\): \(5.5 > \frac{1}{2}(1) + 5\) → \(5.5 > 0.5 + 5\) → \(5.5 > 5.5\) (False; but it's not strictly greater)
- \(y < -2x + 1\): \(5.5 < -2(1) + 1\) → \(5.5 < -2 + 1\) → \(5.5 < -1\) (False)

This point does not satisfy all three inequalities.

Based on the evaluations, the point that satisfies all three inequalities is:

**\((-5, 5)\)**.