To find which point is a solution for the simultaneous inequalities \( y \leq 4x - 5 \) and \( y < 2x - 7 \), we can substitute each point into both inequalities.
Let's check each point:
-
Point (7, 12)
-
For \( y \leq 4x - 5 \):
- \( 12 \leq 4(7) - 5 \)
- \( 12 \leq 28 - 5 \)
- \( 12 \leq 23 \) (True)
-
For \( y < 2x - 7 \):
- \( 12 < 2(7) - 7 \)
- \( 12 < 14 - 7 \)
- \( 12 < 7 \) (False)
-
So (7, 12) does not satisfy both inequalities.
-
-
Point (5, 3)
-
For \( y \leq 4x - 5 \):
- \( 3 \leq 4(5) - 5 \)
- \( 3 \leq 20 - 5 \)
- \( 3 \leq 15 \) (True)
-
For \( y < 2x - 7 \):
- \( 3 < 2(5) - 7 \)
- \( 3 < 10 - 7 \)
- \( 3 < 3 \) (False)
-
So (5, 3) does not satisfy both inequalities.
-
-
Point (0, 15)
-
For \( y \leq 4x - 5 \):
- \( 15 \leq 4(0) - 5 \)
- \( 15 \leq 0 - 5 \)
- \( 15 \leq -5 \) (False)
-
So (0, 15) does not satisfy both inequalities.
-
-
Point (15, 0)
-
For \( y \leq 4x - 5 \):
- \( 0 \leq 4(15) - 5 \)
- \( 0 \leq 60 - 5 \)
- \( 0 \leq 55 \) (True)
-
For \( y < 2x - 7 \):
- \( 0 < 2(15) - 7 \)
- \( 0 < 30 - 7 \)
- \( 0 < 23 \) (True)
-
So (15, 0) satisfies both inequalities.
-
Putting it all together, the only point that satisfies both inequalities is (15, 0).