Question

Как узнать равносильны ли системы уравненмий

Answer 1

How to determine if systems of equations are equivalent

To determine if two systems of equations are equivalent, you need to compare their solutions. If the two systems have the same set of solutions, then they are considered equivalent. Here are the steps to follow:

1. Write down the two systems of equations you want to compare. 2. Solve each system of equations separately to find their solutions. 3. Compare the solutions of the two systems. If the solutions are the same, then the systems are equivalent. If the solutions are different, then the systems are not equivalent.

Let's illustrate this with an example:

Example:

Consider the following systems of equations:

System 1: ``` 2x + 3y = 7 4x - y = 2 ```

System 2: ``` 6x + 9y = 21 12x - 3y = 6 ```

To determine if these systems are equivalent, we need to compare their solutions.

Step 1: Solve each system of equations separately.

For System 1: ``` 2x + 3y = 7 (Equation 1) 4x - y = 2 (Equation 2) ```

Solving Equation 2 for y, we get: ``` y = 4x - 2 ```

Substituting this value of y into Equation 1, we get: ``` 2x + 3(4x - 2) = 7 2x + 12x - 6 = 7 14x = 13 x = 13/14 ```

Substituting the value of x back into Equation 2, we get: ``` 4(13/14) - y = 2 52/14 - y = 2 52 - 14y = 28 -14y = -24 y = 24/14 = 12/7 ```

So the solution to System 1 is x = 13/14 and y = 12/7.

For System 2: ``` 6x + 9y = 21 (Equation 3) 12x - 3y = 6 (Equation 4) ```

Solving Equation 4 for y, we get: ``` y = 4x - 2 ```

Substituting this value of y into Equation 3, we get: ``` 6x + 9(4x - 2) = 21 6x + 36x - 18 = 21 42x = 39 x = 39/42 = 13/14 ```

Substituting the value of x back into Equation 4, we get: ``` 12(13/14) - y = 6 156/14 - y = 6 156 - 14y = 84 -14y = -72 y = 72/14 = 36/7 ```

So the solution to System 2 is x = 13/14 and y = 36/7.

Step 2: Compare the solutions of the two systems.

The solutions to both System 1 and System 2 are x = 13/14 and y = 12/7. Since the solutions are the same, we can conclude that System 1 and System 2 are equivalent.

Therefore, the two systems of equations are equivalent.

Conclusion

To determine if systems of equations are equivalent, you need to compare their solutions. If the solutions are the same, then the systems are equivalent. If the solutions are different, then the systems are not equivalent. In the example provided, we compared the solutions of two systems of equations and found that they were equivalent.