x – 2y = 8; 5x – 10y = 10

Question

In the systems of equation determine whether the system has a unique solution, no solution or infinite solutions. In case there is a unique solution, find it:

x – 2y = 8; 

5x – 10y = 10

Answer 1

The given system of equations is:

x – 2y – 8 = 0

5x – 10y – 10 = 0

The above equations are of the form

a₁ x + b₁ y − c₁ = 0 

a₂ x + b₂ y − c₂ = 0 

Here, a₁ = 1, b₁ = -2, c₁ = −8 

a₂ = 5, b₂ = -10, c₂ = −10

So according to the question, we get 

\(\frac{a_1}{a_2}\) = \(\frac{1}{5}\) 

\(\frac{b_1}{ b_2}\) = \(\frac{-2}{-10}\) = \(\frac{1}{5}\) and, 

\(\frac{c_1}{c_2}\) = \(\frac{−8}{−10}\) = \(\frac{4}{5}\) 

⇒ \(\frac{a_1}{a_2}\) = \(\frac{b_1}{ b_2}\) ≠ \(\frac{c_1}{c_2}\)

Hence, we can conclude that the given system of equation has no solution.