Which of the following points are solutions to the system of inequalities:
4x-y≤6
x+2y<1
Which of the following points are solutions to the system of inequalities:
4x-y≤6
x+2y<1
(0,-2)
(0,6)
(6,0)
(2,0)
(-2,0)
(0,-6)
To determine if a point is a solution to a system of inequalities, we need to substitute the coordinates of the point into each inequality and check if the inequality is true.
Let's check each point one by one:
1) (0,-2)
- Substitute x = 0 and y = -2 into the first inequality: 4(0) - (-2) ≤ 6
Simplify: 2 ≤ 6
The inequality is true.
- Substitute x = 0 and y = -2 into the second inequality: (0) + 2(-2) < 1
Simplify: -4 < 1
The inequality is true.
Therefore, (0,-2) is a solution to the system of inequalities.
2) (0,6)
- Substitute x = 0 and y = 6 into the first inequality: 4(0) - (6) ≤ 6
Simplify: -6 ≤ 6
The inequality is true.
- Substitute x = 0 and y = 6 into the second inequality: (0) + 2(6) < 1
Simplify: 12 < 1
The inequality is false.
Therefore, (0,6) is not a solution to the system of inequalities.
3) (6,0)
- Substitute x = 6 and y = 0 into the first inequality: 4(6) - (0) ≤ 6
Simplify: 24 ≤ 6
The inequality is false.
- Substitute x = 6 and y = 0 into the second inequality: (6) + 2(0) < 1
Simplify: 6 < 1
The inequality is false.
Therefore, (6,0) is not a solution to the system of inequalities.
4) (2,0)
- Substitute x = 2 and y = 0 into the first inequality: 4(2) - (0) ≤ 6
Simplify: 8 ≤ 6
The inequality is false.
- Substitute x = 2 and y = 0 into the second inequality: (2) + 2(0) < 1
Simplify: 2 < 1
The inequality is false.
Therefore, (2,0) is not a solution to the system of inequalities.
5) (-2,0)
- Substitute x = -2 and y = 0 into the first inequality: 4(-2) - (0) ≤ 6
Simplify: -8 ≤ 6
The inequality is true.
- Substitute x = -2 and y = 0 into the second inequality: (-2) + 2(0) < 1
Simplify: -2 < 1
The inequality is true.
Therefore, (-2,0) is a solution to the system of inequalities.
6) (0,-6)
- Substitute x = 0 and y = -6 into the first inequality: 4(0) - (-6) ≤ 6
Simplify: 6 ≤ 6
The inequality is true.
- Substitute x = 0 and y = -6 into the second inequality: (0) + 2(-6) < 1
Simplify: -12 < 1
The inequality is true.
Therefore, (0,-6) is a solution to the system of inequalities.
In conclusion, the solutions to the system of inequalities are:
(0,-2)
(-2,0)
(0,-6)