Question

Нужна помощь, срочно! найдите все решения в целых числах данной системы. {xy+vw=5, {xv-yw=6

Answer 1

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Answer 2

System of Equations

You have provided a system of equations:

1. xy + vw = 5 2. xv - yw = 6

To find the solutions in whole numbers for this system, we can use various methods such as substitution, elimination, or matrix methods. Let's solve this system step by step.

Method 1: Substitution

We can solve the system by isolating one variable in terms of the other and substituting it into the other equation. Let's solve for x in terms of y in the second equation:

xv - yw = 6

xv = yw + 6

x = (yw + 6)/v

Now, substitute this expression for x into the first equation:

xy + vw = 5

((yw + 6)/v)y + vw = 5

Simplifying this equation will give us a quadratic equation in terms of y. Solving this quadratic equation will provide the values of y, and then we can substitute these values back into the expression for x to find the corresponding values.

Method 2: Elimination

Another method to solve this system is by eliminating one variable by adding or subtracting the equations. Let's eliminate the variable x by multiplying the first equation by v and the second equation by w:

v(xy + vw) = 5v

w(xv - yw) = 6w

Expanding these equations will give us:

vxy + v^2w = 5v

wxv - wy^2 = 6w

Now, we can add these equations together to eliminate the variable x:

vxy + wxv + v^2w - wy^2 = 5v + 6w

Factor out the common terms:

x(vy + wv) + vw(v - y) = 5v + 6w

Now, we have an equation in terms of x and y. Solving this equation will give us the values of x and y.

Method 3: Matrix Methods

We can also solve this system using matrix methods, such as Gaussian elimination or matrix inversion. By representing the system of equations as a matrix equation, we can perform row operations to transform the matrix into row-echelon form or reduced row-echelon form. This will allow us to find the solutions for x and y.

Let's represent the system of equations as a matrix equation:

| x  y |   | v  w |   | 5 |
| x -y | * | v -w | = | 6 |

By performing row operations on this matrix equation, we can find the solutions for x and y.

Conclusion

To find the solutions in whole numbers for the given system of equations, we can use methods such as substitution, elimination, or matrix methods. These methods involve manipulating the equations to isolate variables or perform row operations on a matrix representation of the system. The specific solutions will depend on the values of v and w in the system.