Question

ПОМОГИТЕ ПОЖАЛУЙСТА!!!Дам 34 балла!!При каких значениях а система имеет единственное решение? А){3ах+2у=а+3 {6х+4у=2а+1 Б){ах+2у=3а+5 {2х+ау=а-3

Answer 1

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Answer 2

System of Equations

The given system of equations is:

A) 3ax + 2y = a + 3 6x + 4y = 2a + 1

B) ax + 2y = 3a + 5 2x + ay = a - 3

To determine the values of 'a' for which the system has a unique solution, we can use various methods such as substitution, elimination, or matrix operations. Let's solve each system separately to find the values of 'a' that satisfy the condition.

Solving System A

We'll solve system A using the method of elimination. Multiply the first equation by 2 and the second equation by 3 to eliminate 'a':

2(3ax + 2y) = 2(a + 3) 3(6x + 4y) = 3(2a + 1)

Simplifying the equations, we get:

6ax + 4y = 2a + 6 18x + 12y = 6a + 3

Now, subtract the first equation from the second equation:

(18x + 12y) - (6ax + 4y) = (6a + 3) - (2a + 6) 18x + 12y - 6ax - 4y = 6a + 3 - 2a - 6 12x + 8y - 6ax = 4a - 3

Factoring out 'a' on the left side:

a(12x - 6x) + 8y = 4a - 3 6ax + 8y = 4a - 3

Simplifying further:

6ax + 8y = 4a - 3

Now, let's compare this equation with the second equation of system A:

6x + 4y = 2a + 1

We can see that the coefficients of 'x' and 'y' are the same in both equations. This means that the two equations represent the same line. Therefore, system A does not have a unique solution. The values of 'a' do not matter in this case.

Solving System B

We'll solve system B using the method of substitution. Rearrange the first equation to solve for 'x':

ax + 2y = 3a + 5 x = (3a + 5 - 2y) / a

Substitute this value of 'x' into the second equation:

2((3a + 5 - 2y) / a) + ay = a - 3

Simplify the equation:

(6a + 10 - 4y) / a + ay = a - 3

Multiply both sides of the equation by 'a' to eliminate the fraction:

6a + 10 - 4y + a^2y = a^2 - 3a

Rearrange the equation:

a^2y + (6 - 3a)y - 4y = a^2 - 3a - 10

Combine like terms:

(a^2 - 3a + 6 - 4)y = a^2 - 3a - 10

Simplify further:

(a^2 - 3a + 2)y = a^2 - 3a - 10

Now, let's compare this equation with the second equation of system B:

2x + ay = a - 3

We can see that the coefficients of 'x' and 'y' are the same in both equations. This means that the two equations represent the same line. Therefore, system B does not have a unique solution. The values of 'a' do not matter in this case.

Conclusion

In both systems A and B, there are no values of 'a' for which the systems have a unique solution. The systems either have no solution or infinitely many solutions, as the equations represent the same line.