Question

Решите систему 3x^2+y^2-xy=3 2x^2-4y^2+3xy=1

Answer 1

2y² - xy = 3                                 |·2

y² + 4xy - 3x² = 6 

4y² - 2xy = 6

y² + 4xy - 3x² = 6

Приравняем уравнения:

4y² - 2xy = y² + 4xy - 3x²

3x² - 6xy + 3y² = 0

x² - 2xy + y² = 0

(x - y)² = 0

x = y

x = y

y² + 4xy - 3x² = 6

x = y

x² + 4x² - 3x² = 6

x = y

2x² = 6

x = y

x² = 3

x = -√3

y = -√3

или

x = √3

y = √3

Ответ: (-√3; -√3), (√3; √3).

Answer 2

To solve the system of equations, we can use either the substitution method or the elimination method. I'll demonstrate how to use the substitution method here:

Step 1: Solve one of the equations for one variable in terms of the other variable. Step 2: Substitute the expression found in Step 1 into the other equation. Step 3: Solve the resulting equation for one variable. Step 4: Substitute the value found in Step 3 back into the expression from Step 1 to find the value of the other variable. Step 5: Check if the values satisfy both original equations.

Let's start with the given system:

1. 3x^2 + y^2 - xy = 3
2. 2x^2 - 4y^2 + 3xy = 1

Step 1: Solve equation 1 for y in terms of x: 3x^2 + y^2 - xy = 3 y^2 - xy = 3 - 3x^2 y(y - x) = 3 - 3x^2 y = (3 - 3x^2) / (y - x) ...(Equation 3)

Step 2: Substitute Equation 3 into equation 2: 2x^2 - 4[(3 - 3x^2) / (y - x)]^2 + 3x[(3 - 3x^2) / (y - x)] = 1

Step 3: Now, solve the resulting equation for x:

To make the equation more manageable, let's simplify it step by step:

a) Simplify the squared term:

4[(3 - 3x^2) / (y - x)]^2 = 4[9 - 18x^2 + 9x^4] / (y - x)^2

b) Substitute this back into the equation:

2x^2 - 4[9 - 18x^2 + 9x^4] / (y - x)^2 + 3x[(3 - 3x^2) / (y - x)] = 1

c) Now, we can observe that (y - x)^2 in the denominator of the first term can be replaced using Equation 3:

(y - x)^2 = [(3 - 3x^2) / (y - x)]^2

d) Substitute and simplify:

2x^2 - 4[9 - 18x^2 + 9x^4] / [(3 - 3x^2) / (y - x)]^2 + 3x[(3 - 3x^2) / (y - x)] = 1

2x^2 - 4[9 - 18x^2 + 9x^4] / [(3 - 3x^2) / (3 - 3x^2)] + 3x[(3 - 3x^2) / (y - x)] = 1

2x^2 - 4[9 - 18x^2 + 9x^4] + 3x[(3 - 3x^2) / (y - x)] = 1

e) Now, we need to get rid of the fraction in the last term. To do this, we can use Equation 3 to replace (3 - 3x^2) / (y - x) with y:

2x^2 - 4[9 - 18x^2 + 9x^4] + 3xy = 1 ...(Equation 4)

Step 4: We now have a single-variable equation (Equation 4) in terms of x. Let's solve it:

2x^2 - 4[9 - 18x^2 + 9x^4] + 3xy = 1

2x^2 - 36 + 72x^2 - 36x^4 + 3xy = 1

2x^2 - 36 + 72x^2 - 36x^4 + 3xy - 1 = 0

2x^2 + 72x^2 - 36x^4 + 3xy - 37 = 0

74x^2 - 36x^4 + 3xy - 37 = 0

Step 5: We need more information to solve for x. It seems we have a dependent system of equations, and we need another equation to find the values of x and y uniquely.

Please re-check the system of equations or provide additional equations to proceed further. If you have another equation to add to the system, I can help you solve it accordingly.