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:

x − 2y − 8 = 0

5x − 10y − 10 = 0

Answer 1

The given system of equations may be written as:

x − 2y − 8 = 0

5x − 10y − 10 = 0

The given system of equations is of the form,

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Where a₁ = 1, b₁ = −2, c₁ = −8

And a₂ = 5, b₂ = −10, c₂ = −10

We have,

`a_1/a_2 = 1/5`

`b_1/b_2 = (-2)/(-10) = 1/5`

And `c_1/c_2 = (-8)/(-10) = 4/5`

Cleary `a_1/a_2 = b_2/b_2 != c_1/c_2`

So, the given system of equations has no solution.