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:2x + y = 5; 4x + 2y = 10
Answer» The given system of equations is: 2x + y – 5 = 0 4x + 2y – 10 = 0 The above equations are of the form a₁ x + b₁y − c₁ = 0 a₂ x + b₂ y − c₂ = 0 Here, a₁ = 2, b₁ = 1, c₁ = −5 a₂ = 4, b₂ = 2, c₂ = −10 So according to the question, we get \(\frac{a_1}{a_2}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\) \(\frac{b_1}{ b_2}\) = \(\frac{1}{2}\) and, \(\frac{c_1}{c_2}\) = \(\frac{−5}{−10}\) = \(\frac{1}{2}\) ⇒ \(\frac{a_1}{a_2}\) = \(\frac{b_1}{ b_2}\) = \(\frac{c_1}{c_2}\) Hence, we can conclude that the given system of equation has infinity many solutions.

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:2x + y = 5; 4x + 2y = 10

Answer»

The given system of equations is:

2x + y – 5 = 0

4x + 2y – 10 = 0

The above equations are of the form

a₁ x + b₁y − c₁ = 0

a₂ x + b₂ y − c₂ = 0

Here, a₁ = 2, b₁ = 1, c₁ = −5

a₂ = 4, b₂ = 2, c₂ = −10

So according to the question, we get

\(\frac{a_1}{a_2}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\)

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

\(\frac{c_1}{c_2}\) = \(\frac{−5}{−10}\) = \(\frac{1}{2}\)

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

Hence, we can conclude that the given system of equation has infinity many solutions.