Question

Show that the system of equations
$3x-5y=7,6x-10y=3$
is inconsistent.

Answer 1

The given system of equations is:
3x − 5y = 7
⇒ 3x − 5y − 7 = 0 ...(i)
6x − 10y = 3
⇒ 6x − 10y − 3 = 0 ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 3, b₁= −5, c₁ = −7 and a₂ = 6, b₂ = −10, c₂ = −3
∴ $\frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2},\frac{b_1}{b2}=\frac{-5}{-10}=\frac{1}{2}$ and $\frac{c_1}{c_2}=\frac{-7}{-3}=\frac{7}{3}$
Thus, $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
Hence, the given system of equations is inconsistent.