# Condition of a system of equations to have infinitly many solutions

• fluidistic
In summary, the conversation discusses solving a system of equations to find the values of "a" that result in a unique solution or an infinite number of solutions. The individual suggests using Cramer's Rule and mentions that "a" cannot be equal to 1 in the vector solution, but it could be equal to 1 in the original system of equations. They also mention the possibility of making an error and ask for further guidance on finding the values of "a" that satisfy the conditions of the question.
fluidistic
Gold Member
0. Homework Statement
For which values of "a" the following system of equations has a unique solution? Infinitly many solutions?
x-y+z=2
ax-y+z=2
2x-2y+(2-a)z=4a

1. The attempt at a solution

I've put the system of equations under an amplied matrix and I reduced it.
I finally got the 3x3 identity matrix with the corresponding vector solution : { (4-8a+a²)/(1-a) ; (3a-2)/(2a-1)*(-2+6a-4a²)/(1-a)² ; (-2+6a-4a²)/(1-a)² ) }
I don't know what to do from here.
I see that "a" cannot be equal to 1 in the vector solution, but it could be equal to 1 in the original system of equations. So I probably did at least one error. But still, say the result is what I got, what do I have to do to find out the values of "a" satisfying the conditions of the question?

Try using Cramer's Rule. If the denominater determinate equals zero then the solutions are inconsistent or dependent. For an infinite number of solutions, the numerator determinate will equal zero for each variable.

## 1. What is the condition for a system of equations to have infinitely many solutions?

The condition for a system of equations to have infinitely many solutions is that the number of equations must be less than the number of variables in the system. This means that there are more unknowns than equations, allowing for multiple possible solutions.

## 2. Can a system of equations have infinitely many solutions if it has no solutions?

No, a system of equations cannot have infinitely many solutions if it has no solutions. In order for a system to have infinitely many solutions, there must be at least one solution that satisfies all of the equations. If there are no solutions, then there cannot be infinitely many.

## 3. How can I determine if a system of equations has infinitely many solutions?

To determine if a system of equations has infinitely many solutions, you can perform row operations on the augmented matrix of the system. If the resulting matrix has at least one row of all zeros, then the system has infinitely many solutions. This indicates that there are more unknowns than equations, allowing for multiple possible solutions.

## 4. Is it possible for a system of equations to have both infinitely many solutions and no solutions?

No, it is not possible for a system of equations to have both infinitely many solutions and no solutions. If there are infinitely many solutions, then there must be at least one solution that satisfies all of the equations. If there are no solutions, then there cannot be infinitely many solutions.

## 5. What does the graph of a system of equations with infinitely many solutions look like?

The graph of a system of equations with infinitely many solutions will show parallel lines. This is because there is no unique solution to the system, and the lines will never intersect. Instead, they will have the same slope and different y-intercepts, resulting in infinitely many possible solutions.

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