Question

Решить систему3x-y =2x+2y-z=22x-y+z=2

Answer 1

Сначала решал методом Гауса,потом подставил значения из матрицы В систему .Выразил переменные

Answer 2

Решить это в полном смысле нельзя, потому что уравнений 2, а переменных 3. Можно выразить y и z через x, а само x - любое.

{ z = 3 - x - y

{ x^2 + y^2 + (3 - x - y)^2 = 3

Раскрываем скобки

x^2 + y^2 + 9 + x^2 + y^2 - 6x - 6y + 2xy = 3

2x^2 + 2y^2 - 6x - 6y + 2xy + 6 = 0

Делим все на 2

x^2 + y^2 - 3x - 3y + xy + 3 = 0

Распишем, как будто y - переменная, а x - константа.

y^2 + y(x - 3) + (x^2 - 3x + 3) = 0

Решаем как обычное квадратное уравнение

D = (x - 3)^2 - 4(x^2 - 3x + 3) = x^2 - 6x + 9 - 4x^2 + 12x - 12 =

= -3x^2 + 6x - 3 = -3(x^2 - 2x + 1) = -3(x - 1)^2

D < 0 при любом x, кроме 1. При x = 1 будет D = 0

y = (3 - x)/2 = (3 - 1)/2 = 2/2 = 1

z = 3 - x  - y = 3 - 1 - 1 = 1

Решение только одно: (1; 1; 1).

Answer 3

To solve the system of equations, we can use either substitution or elimination method. Let's use the elimination method:

The given system of equations is:

1. 3x - y = 2
2. 2x + 2y - z = 2
3. 2x - y + z = 2

Step 1: We can start by eliminating one variable from two equations. Let's eliminate 'y' from equations 1 and 3:

Multiply equation 1 by 1 and equation 3 by -1:

1. 3x - y = 2
2. -2x + y - z = -2

Adding the above equations: (3x - y) + (-2x + y - z) = 2 - 2 x - z = 0 x = z ...........(4)

Step 2: Now, substitute the value of 'x' from equation (4) into equation 1 or 3 to find the value of 'y.' Let's use equation 1:

3x - y = 2 3(z) - y = 2 y = 3z - 2 ...........(5)

Step 3: Finally, substitute the values of 'x' and 'y' from equations (4) and (5) into equation 2 to find the value of 'z':

2x + 2y - z = 2 2(z) + 2(3z - 2) - z = 2 2z + 6z - 4 - z = 2 7z - 4 = 2 7z = 2 + 4 7z = 6 z = 6/7

Step 4: Now that we have found the value of 'z', we can find the values of 'x' and 'y' using equations (4) and (5):

From equation (4): x = z x = 6/7

From equation (5): y = 3z - 2 y = 3(6/7) - 2 y = 18/7 - 14/7 y = 4/7

So, the solution to the system of equations is: x = 6/7 y = 4/7 z = 6/7