<h2>
Hello!</h2>
The answer is:
The point that is a solution to the system of inequalities is C(4,2).
<h2>
Why?</h2>
To find the point that is a solution to the system of inequalities, we need to evaluate it into the given inequalities. If the point is a solution to the system of inequalities, both inequalities will be satisfied.
We are given the inequalities:
![y\leq 2x-2](https://tex.z-dn.net/?f=y%5Cleq%202x-2)
![y\leq x^{2} -3x](https://tex.z-dn.net/?f=y%5Cleq%20x%5E%7B2%7D%20-3x)
So, substituting the given points into the given inequalities, we have:
- A. (2,1)
Substituting into the first inequality, we have:
![y\leq 2x-2\\\\1\leq 2*(2)-2\\\\1\leq 4-2\\\\1\leq 2](https://tex.z-dn.net/?f=y%5Cleq%202x-2%5C%5C%5C%5C1%5Cleq%202%2A%282%29-2%5C%5C%5C%5C1%5Cleq%204-2%5C%5C%5C%5C1%5Cleq%202)
Substituting into the second inequality, we have:
![y\leq x^{2} -3x](https://tex.z-dn.net/?f=y%5Cleq%20x%5E%7B2%7D%20-3x)
![1\leq (2)^{2} -3(2)](https://tex.z-dn.net/?f=1%5Cleq%20%282%29%5E%7B2%7D%20-3%282%29)
![1\leq (2)^{2} -3(2)](https://tex.z-dn.net/?f=1%5Cleq%20%282%29%5E%7B2%7D%20-3%282%29)
![1\leq 4 -6](https://tex.z-dn.net/?f=1%5Cleq%204%20-6)
![1\leq -2](https://tex.z-dn.net/?f=1%5Cleq%20-2)
Therefore, since 1 is not less or equal to -2, the point A(2,1) is not a solution to the system of inequalities.
- B. (-2,-1)
Substituting into the first inequality, we have:
![y\leq 2x-2\\\\-1\leq (-2)*(2)-2\\\\-1\leq -4-2\\\\-1\leq -6](https://tex.z-dn.net/?f=y%5Cleq%202x-2%5C%5C%5C%5C-1%5Cleq%20%28-2%29%2A%282%29-2%5C%5C%5C%5C-1%5Cleq%20-4-2%5C%5C%5C%5C-1%5Cleq%20-6)
Substituting into the second inequality, we have:
![y\leq x^{2} -3x](https://tex.z-dn.net/?f=y%5Cleq%20x%5E%7B2%7D%20-3x)
![-1\leq (-2)^{2} -3(-2)](https://tex.z-dn.net/?f=-1%5Cleq%20%28-2%29%5E%7B2%7D%20-3%28-2%29)
![-1\leq 4 +6](https://tex.z-dn.net/?f=-1%5Cleq%204%20%2B6)
![-1\leq 10](https://tex.z-dn.net/?f=-1%5Cleq%2010)
Therefore, since -1 is not less or equal to -6, the point B(-2,-1) is not a solution to the system of inequalities.
- C. (4,2)
Substituting into the first inequality, we have:
![y\leq 2x-2\\\\2\leq (4)*(2)-2\\\\2\leq 8-2\\\\2\leq 6](https://tex.z-dn.net/?f=y%5Cleq%202x-2%5C%5C%5C%5C2%5Cleq%20%284%29%2A%282%29-2%5C%5C%5C%5C2%5Cleq%208-2%5C%5C%5C%5C2%5Cleq%206)
Substituting into the second inequality, we have:
![y\leq x^{2} -3x](https://tex.z-dn.net/?f=y%5Cleq%20x%5E%7B2%7D%20-3x)
![2\leq (4)^{2} -3(4)](https://tex.z-dn.net/?f=2%5Cleq%20%284%29%5E%7B2%7D%20-3%284%29)
![2\leq 16 -12](https://tex.z-dn.net/?f=2%5Cleq%2016%20-12)
![2\leq 4](https://tex.z-dn.net/?f=2%5Cleq%204)
Therefore, since 2 is less than 6, and 2 is less than 4, the point B(4,2) is a solution to the system of inequalities.
- D(1,3)
Substituting into the first inequality, we have:
![y\leq 2x-2\\\\3\leq (1)*(2)-2\\\\3\leq 2-2\\\\3\leq 0](https://tex.z-dn.net/?f=y%5Cleq%202x-2%5C%5C%5C%5C3%5Cleq%20%281%29%2A%282%29-2%5C%5C%5C%5C3%5Cleq%202-2%5C%5C%5C%5C3%5Cleq%200)
Substituting into the second inequality, we have:
![y\leq x^{2} -3x](https://tex.z-dn.net/?f=y%5Cleq%20x%5E%7B2%7D%20-3x)
![3\leq (1)^{2} -3(1)](https://tex.z-dn.net/?f=3%5Cleq%20%281%29%5E%7B2%7D%20-3%281%29)
![3\leq 1 -3](https://tex.z-dn.net/?f=3%5Cleq%201%20-3)
![3\leq -2](https://tex.z-dn.net/?f=3%5Cleq%20-2)
Therefore, since 3 is not less than 0, and 3 is not less than -2, the point D(1.3) is not a solution to the system of inequalities.
Hence, the point that is a solution to the system of inequalities is C(4,2).
Have a nice day!