Question

Решить систему уравнений. Пожалуйста подробно!

Answer 1

Logx(y) = 1\logy(x)
Делаем замену, logy(x) = z
z + 1\z = 2
z^2 + 1 = 2z
z^2 - 2z + 1 = 0
z1 + z2 = 2
z1*z2 = 1
z = 1

logy(x) = 1
y = 12 - x^2
y = x

y^2 + y - 12 = 0
y1 + y2 = -1
y1*y2 = -12
y1 = -4
y2 = 3

Из определения логарифма имеем ограничение на x и y, они должны быть строго больше нуля.
Поэтому x = y = 3

Answer 2

Solving a System of Equations

To solve a system of equations, we need to find the values of the variables that satisfy all the equations in the system. Let's consider the examples you provided and solve them step by step.

# Example 1: 20x + 5 = -20, 9x + 2 = 25

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

From the first equation, we have:

20x + 5 = -20

Subtracting 5 from both sides:

20x = -25

Dividing both sides by 20:

x = -25/20 = -5/4

Now, let's substitute this value of x into the second equation:

9(-5/4) + 2 = 25

Simplifying:

-45/4 + 2 = 25

Combining like terms:

-45/4 + 8/4 = 25

-37/4 = 25

To solve for x, we can multiply both sides by 4:

-37 = 100

This equation is not true, which means there is no solution that satisfies both equations. Therefore, this system of equations is inconsistent and has no solution.

# Example 2: 3x + 2y = 3, 3x - 4y = 18

Let's solve this system of equations using the method of elimination.

First, let's multiply the first equation by 2 and the second equation by 3 to eliminate the x term:

2(3x + 2y) = 2(3) 3(3x - 4y) = 3(18)

Simplifying:

6x + 4y = 6 9x - 12y = 54

Now, let's multiply the first equation by 3 and the second equation by 2 to eliminate the y term:

3(6x + 4y) = 3(6) 2(9x - 12y) = 2(54)

Simplifying:

18x + 12y = 18 18x - 24y = 108

Now, let's subtract the second equation from the first equation to eliminate the x term:

(18x + 12y) - (18x - 24y) = 18 - 108

Simplifying:

18x + 12y - 18x + 24y = -90

Combining like terms:

36y = -90

Dividing both sides by 36:

y = -90/36 = -5/2

Now, let's substitute this value of y into the first equation:

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

Simplifying:

3x - 5 = 3

Adding 5 to both sides:

3x = 8

Dividing both sides by 3:

x = 8/3

Therefore, the solution to this system of equations is x = 8/3 and y = -5/2.

Conclusion

In summary, we solved two examples of systems of equations. The first example was inconsistent and had no solution, while the second example had a unique solution with x = 8/3 and y = -5/2.