Question

Show that the equations $9x-10y=21,\frac{3x}{2}-\frac{5y}{3}=\frac{7}{2}$ have infinitely many solutions.

Answer 1

The given system of equations can be written as follows:
$9x-10y-21=0\mathrm{and}\frac{3x}{2}-\frac{5y}{3}-\frac{7}{2}=0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 9, b₁ = −10, c₁ = −21 and a₂ = $\frac{3}{2}$, b₂ = $\frac{-5}{3}$ and c₂ = $\frac{-7}{2}$
∴ $\frac{a_1}{a_2}=\frac{9}{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=6,\frac{b_1}{b_2}=\frac{-10}{\left({\displaystyle \raisebox{1ex}{$-5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}=6\mathrm{and}\frac{c_1}{c_2}=-21\times \frac{2}{-7}=6$
∴ $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
This shows that the given system equations has an infinite number of solutions.