Question

How many solutions are there for the system shown below?
2x² + y2 = 33
x2 + y + 2y=19

Answer 1

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.