Linear System Solutions for a Unique Value of a | Math 115 Homework

[JettyZ]

Homework Statement

I'm trying to solve #1 here:
http://www.student.math.uwaterloo.ca/~math115/Exams/M115.FE.pdf

The problem is:

Consider the linear system:
x + ay = 1
ax + 4y = 2

(a) For what values of a does the system have a unique solution?
(b) For what values of a does the system have infinitely many solutions?
(c) For what values of a does the system have no solution?
(d) For a = 1 find the general solution.

The Attempt at a Solution

(a) At a=2, the slope is the same. I figure any value but 2 gives unique solutions. But I don't know if this is right.
(b) At a=2, the slopes are the same, but lines are different. I cannot find any value of a that would give the same slope and same outputs for inputs of x. DNE is the answer?
(c) At a=2, there are no solutions for the system because the lines are parallel to each other. a=2 is the answer.
(d) I think this part was done correctly. I used substitution and solved for the x and y values at a=1 and resulted with (2/3, 1/3)

I would like to know about a, b, and c. I don't know if I did these right.

Thanks.

 

[eumyang]

Ignoring (a) and (b) for the moment:

(a) At a=2, the slope is the same. I figure any value but 2 gives unique solutions. But I don't know if this is right.
(b) At a=2, the slopes are the same, but lines are different. I cannot find any value of a that would give the same slope and same outputs for inputs of x. DNE is the answer?
(c) At a=2, there are no solutions for the system because the lines are parallel to each other. a=2 is the answer.
(d) I think this part was done correctly. I used substitution and solved for the x and y values at a=1 and resulted with (2/3, 1/3)

(d) is right, but (c) is wrong. At a = 2, you have this system:
x + 2y = 1
2x + 4y = 2
Parallel lines have the same slope, but different y-intercepts. Is that the case here?

 

[chiro]

Hey JettyZ.

For c), there are many solutions corresponding to x = t and y = 1 - 2t for any value of t.
For a) and b) you need to use determinants and check for possibilities of inconsistent solutions.

 

[HallsofIvy]

b) At a=2, the slopes are the same, but lines are different. I cannot find any value of a that would give the same slope and same outputs for inputs of x.

If a= 2, your equations are x+ 2y= 1 and 2x+ 4y= 2. If you multiply the first equation by 2, what happens? What does that tell you?

 

[Ray Vickson]

Homework Statement

I'm trying to solve #1 here:
http://www.student.math.uwaterloo.ca/~math115/Exams/M115.FE.pdf

The problem is:

The Attempt at a Solution

(a) At a=2, the slope is the same. I figure any value but 2 gives unique solutions. But I don't know if this is right.
(b) At a=2, the slopes are the same, but lines are different. I cannot find any value of a that would give the same slope and same outputs for inputs of x. DNE is the answer?
(c) At a=2, there are no solutions for the system because the lines are parallel to each other. a=2 is the answer.
(d) I think this part was done correctly. I used substitution and solved for the x and y values at a=1 and resulted with (2/3, 1/3)

I would like to know about a, b, and c. I don't know if I did these right.

Thanks.

Do you know about determinants and their relationship to such questions? If so, use a determinant; you will see that there is a critical value of 'a' that you have missed.

RGV