To find the solution to the given system of equations, we can use the method of substitution or elimination. Let's solve it using the substitution method.
We are given two equations:
1) Y = 2x^2
2) y = -3x - 1
To solve this system, we need to find the values of x and y that satisfy both equations simultaneously.
First, let's substitute equation 2) into equation 1) to eliminate one variable. We substitute y in equation 1) with -3x - 1:
Y = 2x^2
-3x - 1 = 2x^2
Now we have a quadratic equation. To solve it, we can rearrange it into standard form:
2x^2 + 3x + 1 = 0
To factorize this quadratic equation, we need to find two numbers whose product is equal to (2 * 1) = 2 and whose sum is equal to (3 * 1) = 3. The numbers that satisfy these conditions are 1 and 2:
(2x + 1)(x + 1) = 0
Setting each factor equal to zero gives us two possible solutions:
2x + 1 = 0 or x + 1 = 0
Solving these equations for x gives us:
2x = -1 or x = -1
Dividing both sides of the first equation by 2, we get:
x = -1/2
Now that we have found the value of x, we can substitute it back into either equation to find the corresponding value of y. Let's substitute it into equation 2):
y = -3(-1/2) - 1
y = (3/2) - (2/2)
y = (3/2) - (1/2)
y = 2/2
y = 1
Therefore, the solution to the given system of equations is x = -1/2 and y = 1.