Show that the system of equations 2x – 3y = 5, 6x – 9y = 15 has an

1.	Show that the system of equations 2x – 3y = 5, 6x – 9y = 15 has an infinite number of solutions.
Answer» Given system of equations are 2x – 3y = 5 and 6x – 9y = 15. Now, comparing given system of equations with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, We get a₁ = 2, b₁ = – 3 c₁= – 5 and a₂ = 6, b₂ = – 9, c₂ = – 15. Now, \(\frac{a_1}{a_2}\) = \(\frac{2}{6} = \frac{1}{3}\) , \(\frac{b_1}{b_2} = \frac{-3}{-9}\) = \(\frac{1}{3}\) and \(\frac{c_1}{c_2} = \frac{-5}{-15} = \frac{1}{3}\). Hence, \(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\)= \(\frac{c_1}{c_2}\) which is condition of infinite many solutions . Hence, the given system of equations has an infinite number of solutions.

Show that the system of equations 2x – 3y = 5, 6x – 9y = 15 has an infinite number of solutions.

Answer»

Given system of equations are 2x – 3y = 5 and 6x – 9y = 15.

Now, comparing given system of equations with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0,

We get a₁ = 2, b₁ = – 3 c₁= – 5 and a₂ = 6, b₂ = – 9, c₂ = – 15.

Now, \(\frac{a_1}{a_2}\) = \(\frac{2}{6} = \frac{1}{3}\) , \(\frac{b_1}{b_2} = \frac{-3}{-9}\) = \(\frac{1}{3}\) and \(\frac{c_1}{c_2} = \frac{-5}{-15} = \frac{1}{3}\).

Hence, \(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\)= \(\frac{c_1}{c_2}\) which is condition of infinite many solutions .

Hence, the given system of equations has an infinite number of solutions.

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