# One solution, no solution or infinite solutions with k

1. May 1, 2016

### Mrencko

1. The problem statement, all variables and given/known data
¿which one values of "k" the next equation sistem dont have solutions, exactly one solution, infinite solutions?
2. Relevant equations

3. The attempt at a solution
well i guess i need to assign values to test the equations, but i am asking for a metod to help solve this if they are more complex, by the way i am finding solutions, like k=6
but i dont know if foward in number scale are there other solutions

2. May 1, 2016

### SammyS

Staff Emeritus
In general, how do you solve such a system for x and y ?

3. May 1, 2016

### Mrencko

well doing matrix, or for this simple equations sustitution metod, but how to solve whith a teorical k in place?

4. May 1, 2016

### SammyS

Staff Emeritus
Leave it as k. Try substitution.
Just keep k and solve as you would normally solve with substitution.

5. May 1, 2016

### Mrencko

do you mean, solve like k doesnt exist? and put a 0?

6. May 1, 2016

### SammyS

Staff Emeritus
No leave it as k.
Use algebra.

7. May 1, 2016

### Mrencko

its says k=6 so i am ok, but its ok to do the metod whith complex equations? like 5x5

8. May 1, 2016

### SammyS

Staff Emeritus
Now put 6 in for k. Solve the system again in this specific case. (Matrix method may be clearer.)

(I don't understand your last question.)

9. May 1, 2016

### Mrencko

there are some exercices when they put 3x3 equation sistem and in all the equations a k, or in two of them and they ask the same about solutions

10. May 1, 2016

### SammyS

Staff Emeritus
If you use a 6 for k, how would you describe the solution set?

11. May 1, 2016

### Mrencko

correct me if i am wrong but must be k=6 and must be x>y because x-y=positive number, so y=x-3

12. May 1, 2016

### SammyS

Staff Emeritus
k must be 6 , that's correct.

y does not have to be positive.

It is also correct that y = x - 3.

How do you describe the equation y = x - 3 ?

How many possible solutions does that give for (x,y) ?

13. May 1, 2016

### epenguin

Maybe it's unfair to call this 'obvious'. Maybe it's obviousness depends on familiarity.

But if it doesn't hit you in the eye now, after you divide the second equation by 2 doesn't it?

14. May 1, 2016

### Mrencko

I think its a linear function, starts at - 3 in y axis, and the solutions for that only and specific equation must be infinite

15. May 1, 2016

### Mrencko

Yes this is easy, i am looking for a metod to do this right and quick for more complex equation system, for my test

16. May 1, 2016

### SammyS

Staff Emeritus

The y-intercept is -3, but the line doesn't start there.

17. May 1, 2016

### Staff: Mentor

The system is
$$x - y = 3$$
$$2x - 2y = k$$
It might be helpful to look at the geometry here. If k = 6, the two equations are equivalent -- they describe exactly the same line. Any point (x, y) on one line is also on the other line.

If $k \ne 6$, what can you say about the lines? Do they intersect?

Is it possible for the two lines to intersect at exactly one point?

Last edited: May 2, 2016
18. May 1, 2016

### Mrencko

interesting, the way of the geometry, well if k its not 6 then the lines will be parallel whiout any intersection, am i right?

19. May 1, 2016

### SammyS

Staff Emeritus
Right !

20. May 1, 2016

### Mrencko

thanks for your time, people it was very rich the explanation and hit direct into the point of my question, thanks really