Question

Explain why a system of three equations in three variables can have infinitely many solutions

Answer 1

we know that

we will get solution of three system of equations when intersects at one point

For example:

x+2y-3z=-3

2x-5y+4z=13

5x+4y-z=5

These three system of equations has one solutions

so, it has unique solution

that is (2, -1 , 1)

And when planes are lying on one another

then we can get infinite solutions

For example:

x+2y-3z=-3

2x+4y-6z=-6

5x+4y-z=5

So, we will get value of x and y in terms of z

and z can be any value

so, x and y can also be any values

that's why there would be infinite number of solutions