Final answer:
There are two solutions for the given system of equations.
Step-by-step explanation:
The given system of equations is:
2x² + y² = 33
x² + y + 2y = 19
To find the number of solutions, we need to solve the system of equations. Let's start by solving the second equation for x in terms of y:
x² + y + 2y = 19
x² + 3y = 19
x² = 19 - 3y
x = √(19 - 3y)
Now substitute this value of x into the first equation:
2(√(19 - 3y))² + y² = 33
2(19 - 3y) + y² = 33
38 - 6y + y² = 33
y² - 6y + 5 = 0
This is a quadratic equation in y. Using the quadratic formula, we can find the solutions:
y = (-(-6) ± √((-6)² - 4(1)(5))) / (2(1))
y = (6 ± √(36 - 20)) / 2
y = (6 ± √16) / 2
y = (6 ± 4) / 2
y = 5 or y = 1
So, there are two solutions for the given system of equations.