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 − 3y = 3; 3x − 9y = 2
Answer» The given system of equations is: x − 3y – 3 = 0 3x − 9y − 2 = 0 The above equations are of the form a₁ x + b₁ y − c₁ = 0 a₂ x + b₂ y − c₂ = 0 Here, a₁ = 1, b₁ = −3, c₁ = −3 a₂ = 3, b₂ = −9, c₂ = −2 So according to the question, we get \(\frac{a_1}{a_2}\) = \(\frac{1}{3}\) \(\frac{b_1}{b_2}\) = \(\frac{−3}{−9}\) = \(\frac{1}{3}\) and, \(\frac{c_1}{c_2}\) = \(\frac{−3}{−2}\) = \(\frac{3}{2}\) ⇒ \(\frac{a_1}{a_2}\) =\(\frac{b_1}{b_2}\) ≠ \(\frac{c_1}{c_2}\) 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 − 3y = 3; 3x − 9y = 2

Answer»

The given system of equations is:

x − 3y – 3 = 0

3x − 9y − 2 = 0

The above equations are of the form

a₁ x + b₁ y − c₁ = 0

a₂ x + b₂ y − c₂ = 0

Here, a₁ = 1, b₁ = −3, c₁ = −3

a₂ = 3, b₂ = −9, c₂ = −2

So according to the question, we get

\(\frac{a_1}{a_2}\) = \(\frac{1}{3}\)

\(\frac{b_1}{b_2}\) = \(\frac{−3}{−9}\) = \(\frac{1}{3}\) and,

\(\frac{c_1}{c_2}\) = \(\frac{−3}{−2}\) = \(\frac{3}{2}\)

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

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