Question

Решить систему уравнений х2-2y2+4xy=-45 y2-xy-6x2=0 пожалуйста, объясните как решить! спасибо

Answer 1

x^2-2y^2+4*x*y=-45

y^2-x*y-6x^2=0

 

Второе уравнение решаем относительно Y

 

D=b^2-4ac=x-4*(-6x)=25x^2

y1,2=(x±sqrt(25x^2))/2

y1=(x+5x)/2=3x

y2=(x-5x)/2=-2x

Подставляем значение y в первое уравнение

1)  y=3x

    x^2-2*(3x)^2+4*x*3x=-45

    x^2-18x^2+12x^2=-45

    -5x^2=-45

    x^2=9

    x1=3

    x2=-3

     и соответственно

    y=3x

    y1=9

    y2=-9

 

2) y=-2x

     x^2-2*(-2x)^2+4*x*(-2x)=-45

     x^2-8x^2-8x^2=-45

     x^2=-45 -  нет решений

ответ:

   x1=3  

   y1=9

 

   x2=-3

   y2=-9

Answer 2

System of Equations

To solve the system of equations:

1. Equation 1: x^2 - 2y^2 + 4xy = -45 2. Equation 2: y^2 - xy - 6x^2 = 0

We can use various methods, such as substitution or elimination, to find the solutions. Let's solve it using the substitution method.

Substitution Method

1. Solve Equation 2 for y in terms of x: y^2 - xy - 6x^2 = 0 y^2 = xy + 6x^2 y = sqrt(xy + 6x^2) (Equation 3)

2. Substitute Equation 3 into Equation 1: x^2 - 2(sqrt(xy + 6x^2))^2 + 4x(sqrt(xy + 6x^2)) = -45

Simplifying the equation: x^2 - 2(xy + 6x^2) + 4x(sqrt(xy + 6x^2)) = -45 x^2 - 2xy - 12x^2 + 4x(sqrt(xy + 6x^2)) = -45 -11x^2 + (4x(sqrt(xy + 6x^2))) - 2xy = -45

Rearranging the equation: 4x(sqrt(xy + 6x^2)) = 11x^2 - 2xy - 45 4(sqrt(xy + 6x^2)) = 11x - (2y/x) - (45/x) (Equation 4)

3. Now we have two equations: - Equation 3: y = sqrt(xy + 6x^2) - Equation 4: 4(sqrt(xy + 6x^2)) = 11x - (2y/x) - (45/x)

We can substitute Equation 3 into Equation 4 to solve for x.

4. Substitute y from Equation 3 into Equation 4: 4(sqrt(x(sqrt(xy + 6x^2)) + 6x^2)) = 11x - (2(sqrt(xy + 6x^2))/x) - (45/x)

Simplifying the equation: 4(sqrt(x(sqrt(xy + 6x^2)) + 6x^2)) = (11x^2 - 2(sqrt(xy + 6x^2))) - (45/x)

Squaring both sides of the equation: 16(x(sqrt(xy + 6x^2)) + 6x^2) = (11x^2 - 2(sqrt(xy + 6x^2)))^2 - (90/x) + 45

Expanding and simplifying the equation: 16x(sqrt(xy + 6x^2)) + 96x^2 = 121x^4 - 44x^2(sqrt(xy + 6x^2)) + 4(xy + 6x^2) - (90/x) + 45

Rearranging the equation: 121x^4 - 60x^2(sqrt(xy + 6x^2)) + 4(xy + 6x^2) - 16x(sqrt(xy + 6x^2)) - 96x^2 + (90/x) - 45 = 0

5. Now we have a single equation in terms of x. We can solve this equation to find the values of x.

It is important to note that solving this equation analytically may be quite complex and time-consuming. It might be more efficient to use numerical methods or software to find the solutions.

Please let me know if you would like further assistance or if you have any other questions.