# One solution, no solution or infinite solutions with k

## Homework Statement

¿which one values of "k" the next equation sistem dont have solutions, exactly one solution, infinite solutions?

## 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
[/B]

SammyS
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## Homework Statement

¿which one values of "k" the next equation sistem dont have solutions, exactly one solution, infinite solutions?

## Homework Equations

View attachment 100081

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

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

SammyS
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well doing matrix, or for this simple equations sustitution metod, but how to solve whith a teorical k in place?
Leave it as k. Try substitution.
well doing matrix, or for this simple equations sustitution metod, but how to solve whith a teorical k in place?
Just keep k and solve as you would normally solve with substitution.

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

SammyS
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do you mean, solve like k doesnt exist? and put a 0?
No leave it as k.
Use algebra.

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

SammyS
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its says k=6 so i am ok, ...

but its ok to do the metod whith complex equations? like 5x5
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.)

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

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

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

SammyS
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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
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) ?

epenguin
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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?

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

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

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

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

Mark44
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## Homework Statement

¿which one values of "k" the next equation sistem dont have solutions, exactly one solution, infinite solutions?

## Homework Equations

View attachment 100081

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

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

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

so the system have infinite solutions, as long the functions overlap

SammyS
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so the system have infinite solutions, as long the functions overlap
What if the overlap is just a single point?

then there should be just one solution, because there isnt two functions, that overlap in one point and then overlap eachother to ifinite

SammyS
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then there should be just one solution, because there isnt two functions, that overlap in one point and then overlap eachother to ifinite
What you say is true for linear functions, which is probably all that you are considering.

At some point you asked about more complicated systems such as 3×3 or 5×5 .

Those can best be analyzed using matrix methods.

Mark44
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so the system have infinite solutions, as long the functions overlap

What if the overlap is just a single point?

then there should be just one solution, because there isnt two functions, that overlap in one point and then overlap eachother to ifinite
Can the system in this thread have just a single solution? In other words, can the lines represented by the two equations of this system intersect at exactly one point?