Write as a Vector Equality 5x^2+3xy-5y^2=-9 , x^2-xy-y^2=-5

Problem

{[5*x^2+3*x*y−5*y^2=−9],[x^2−x*y−y^2=−5])

Solution

1. Identify the quadratic form for each equation. A quadratic form a*x^2+b*x*y+c*y^2 can be written as x^T*A*x where x=([x],[y]) and A is a symmetric matrix.

2. Construct the symmetric matrix (A_1) for the first equation 5*x^2+3*x*y−5*y^2 The diagonal entries are the coefficients of x^2 and y^2 and the off-diagonal entries are half the coefficient of x*y

(A_1)=([5,1.5],[1.5,−5])

3. Construct the symmetric matrix (A_2) for the second equation x^2−x*y−y^2

(A_2)=([1,−0.5],[−0.5,−1])

4. Express the system as a single vector equality by defining the vector of constants b

b=([−9],[−5])

5. Combine the quadratic forms into a mapping ƒ(x) that equals the constant vector.

Final Answer

([x^T*([5,1.5],[1.5,−5])*x],[x^T*([1,−0.5],[−0.5,−1])*x])=([−9],[−5])

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