Question

In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

2x + y - 5 = 0

4x + 2y - 10 = 0

Answer 1

The given system of equation may be written as

2x + y - 5 = 0

4x + 2y - 10 = 0

The given system of equations is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 2, b_1 = 1, c_1 = -5`

And `a_2 = 4, b_2 = 2, c_2 = -10`

We have

`a_1/a_2 = 2/4 = 1/2`

`b_1/b_2 = 1/2`

And `c_1/c_2 = (-5)/(-10) = 1/2`

cleary `a_1/a_2 = b_1/b_2 = c_1/c_2`

So, the given system of equation has infinity many solutions.