Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution. 2x-y=5 and 4x+ky=2

The system of linear equations has infinitely many solutions

Step-by-step explanation:

Let's modified the equations and find the answer.

Using the first equation:

we can multiply by 2 in both sides, obtaining:

which can by simplified as:

which is equal to:

Considering the second equation:

Taking into account that from the first equation we know that: , we can express the second equation as:

, which can be simplified as:

Because (-8) is being divided by (2+k), then (2+k) can't be equal to 0, so:

if

This means that k can be any number different than -2, and for each of these solutions, there is a different solution for y, allowing also, different solutions for x.

For example, if k=0 then

which give us y=-4, and, because:

if y=-4 then

Now let's try with k=-1, then:

which give us y=-8, and, because:

if y=-8 then .

Then, the system of linear equations has infinitely many solutions

x=12 units bd= 10 units

step-by-step explanation:

i hope this is what you were looking for.

see attached graph