In the systems of equation determine whether the system has a unique s

1.	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» The given system of equations is: x – 2y – 8 = 0 5x – 10y – 10 = 0 The above equations are of the form a₁ x + b₁y − 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.

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»

The given system of equations is:

x – 2y – 8 = 0

5x – 10y – 10 = 0

The above equations are of the form

a₁ x + b₁y − 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.