Solve the system by elimination.
Y=x^2+2,
-4x-y=10
To solve the system of equations by elimination, we'll need to manipulate the equations so that the coefficients of one of the variables will cancel out when the equations are added or subtracted together.
Let's start by solving one of the equations for one variable in terms of the other variable.
In the first equation, Y = x^2 + 2. Let's rearrange the equation to isolate Y:
Y - 2 = x^2
Y = x^2 + 2
Now, we can substitute this expression for Y into the second equation, -4x - y = 10:
-4x - (x^2 + 2) = 10
Expanding the parentheses, we get:
-4x - x^2 - 2 = 10
Rearranging this equation, we have:
x^2 + 4x + 12 = 0
Now, we have a quadratic equation in terms of x. We can solve this equation by factoring or by using the quadratic formula. Factoring gives us:
(x + 2)(x + 6) = 0
Setting each factor equal to zero, we get:
x + 2 = 0 or x + 6 = 0
Solving these equations, we find two possible values for x:
x = -2 or x = -6
Now, we substitute these values of x back into one of the original equations to find the corresponding values of Y.
When x = -2:
Y = (-2)^2 + 2 = 4 + 2 = 6
When x = -6:
Y = (-6)^2 + 2 = 36 + 2 = 38
Therefore, the solution to the system of equations is:
(x, Y) = (-2, 6) and (-6, 38)