systems of nonlinear equations
i need to help i have tried this problem but i cannot find the right answer i need to determine the real solutions for this:
x^2+4xy=10
4x^2-xy=6
To determine the real solutions for the given system of equations:
1. Start by rearranging the equations:
Equation 1: x^2 + 4xy = 10
Equation 2: 4x^2 - xy = 6
2. Solve one of the equations (Equation 1 or Equation 2) for one variable in terms of the other. Let's solve Equation 1 for x in terms of y:
x^2 + 4xy = 10
x^2 = 10 - 4xy
x = sqrt(10 - 4xy)
3. Substitute this expression for x into the other equation (Equation 2):
4(sqrt(10 - 4xy))^2 - xy = 6
4(10 - 4xy) - xy = 6
40 - 16xy - xy = 6
40 - xy(16 + 1) = 6
40 - 17xy = 6
-17xy = 6 - 40
-17xy = -34
xy = 2
4. Substitute the value of xy (found in step 3) back into either of the original equations. Let's use Equation 1:
x^2 + 4(2) = 10
x^2 + 8 = 10
x^2 = 10 - 8
x^2 = 2
x = sqrt(2)
x ≈ ±1.414
Therefore, the real solutions to the system of equations are approximately x ≈ ±1.414 and xy = 2.