Determine all (a, b) so system |x|+y=a, 2|y|-x=b has exactly 3 solns.

[ondri22]

Hello, i need help with my homework, can u please help me?
Determine all pairs of real parameters (a,b) for which the system of equations has exactly three solutions in real numbers.
|x|+y=a
2|y|-x=b
Please help, i´m helpless. (Sorry for my english)

 

[ondri22]

equations

Hello. Can you please help me? I am helpless. Thank you.
Determine all pairs of the real parameters (a,b) for which has a system of equations |x|+y=a and 2|y|-x=b exactly three solutions in real numbers. Sorry for my english.

 

[stapel]

Determine all pairs of the real parameters (a,b) for which has a system of equations |x|+y=a and 2|y|-x=b exactly three solutions in real numbers.

A good first start might be to break apart the absolute values. For instance:

. . . . .|x| + y = a

Then:

. . . . .|x| = a - y

If x > 0, then there is no sign change when the bars are removed, so:

. . . . .x = a - y

However, if x < 0, then there is a sign change, so:

. . . . .x = -(a - y) = y - a

And so forth.

By the way, this is a system of two equations in four unknowns. What techniques have you been given for solving this sort of thing?

Please be complete. Thank you!

 

[ondri22]

Thank you, a and b are parameters and we are supposed to find all possible pairs of these parameters (a,b) for which the system of equation has 3 solutions.

 

[ondri22]

A good first start might be to break apart the absolute values. For instance:

. . . . .|x| + y = a

Then:

. . . . .|x| = a - y

If x > 0, then there is no sign change when the bars are removed, so:

. . . . .x = a - y

However, if x < 0, then there is a sign change, so:

. . . . .x = -(a - y) = y - a

And so forth.

By the way, this is a system of two equations in four unknowns. What techniques have you been given for solving this sort of thing?

Please be complete. Thank you!

Thank you. Actually there are two unknowns x and y. a and b are parameters and we are supposed to determine all pairs of these parameters (a,b) for which will this system have 3 solutions.

 

[Steven G]

Hello. Can you please help me? I am helpless. Thank you.
Determine all pairs of the real parameters (a,b) for which has a system of equations |x|+y=a and 2|y|-x=b exactly three solutions in real numbers. Sorry for my english.

There are 4 cases to consider.
1) x>0 and y>0
2) x>0 and y<0
3) x<0 and y>0
4) x<0 and y<0

Let's look at case 1
|x|+y=a and 2|y|-x=b which become x+y=a and 2y - x = b
So 3y = a+b
Continue...

 

[ondri22]

There are 4 cases to consider.
1) x>0 and y>0
2) x>0 and y<0
3) x<0 and y>0
4) x<0 and y<0

Let's look at case 1
|x|+y=a and 2|y|-x=b which become x+y=a and 2y - x = b
So 3y a+b
Continue...

Thank you very much! Can I have a question? I understand what you did with these equations, but what am I supposed to do then?

 

[stapel]

Thank you very much! Can I have a question? I understand what you did with these equations, but what am I supposed to do then?

The volunteer explained what to do, gave you the cases, and did one of the cases for you. Now you do the same process for the other three cases.

 

[ondri22]

The volunteer explained what to do, gave you the cases, and did one of the cases for you. Now you do the same process for the other three cases.

Thank you for your answer. I know, i did it with other cases bud what then? Don-t know what to do

 

[stapel]

Thank you for your answer. I know, i did it with other cases bud what then? Don-t know what to do

You've solved the cases, getting the various solutions for the system of equations. You've found the various values of a and b. You were asked for the various values of a and b. Where are you "stuck", exactly?

When you reply, please show your work. Thank you!

 

[ondri22]

You've solved the cases, getting the various solutions for the system of equations. You've found the various values of a and b. You were asked for the various values of a and b. Where are you "stuck", exactly?

When you reply, please show your work. Thank you!

1)3y=a+b
2)x+y=a and-2y-x=b so -y=a+b
3)-x+y=a and 2y-x=b so y=b-a
4) -x+y=a and -2y-x=b so -3y=b-a
We are asked to find values of (a,b) so that the system has exactly three solutions. What am I supposed to do now? Solve the system of these three equations? Still don't know how will this lead me to the answer. Thank you a lot for your help!! You're the best.

 

[stapel]

1)3y=a+b
2)x+y=a and-2y-x=b so -y=a+b
3)-x+y=a and 2y-x=b so y=b-a
4) -x+y=a and -2y-x=b so -3y=b-a
We are asked to find values of (a,b) so that the system has exactly three solutions. What am I supposed to do now? Solve the system of these three equations? Still don't know how will this lead me to the answer. Thank you a lot for your help!! You're the best.

Don't just look at the exercise. Do something! Start with the instructions you've been provided here, and do the steps, as you were shown for the first case.

Then, if you're not sure what to do with your results, please reply showing your steps and results, so we can see where you're getting stuck. Thank you!