Is the pair of equations $3 x - 5 y = 6$ and $4 x - 6 y = 7$ consistent $?$ Justify your answer.

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Solution

Given system of equations are
$3 x - 5 y - 6 = 0 . . . . (1)$
$4 x - 6 y - 7 = 0 . . . . (2)$
These given equations are in the form:
$a_{1} x + b_{1} y + c_{1} = 0$ and
$a_{2} x + b_{2} y + c_{2} = 0$
Where
$\begin{aligned} a_{1} = 3, b_{1} = - 5 and c_{1} = - 6 \end{aligned}$
$a_{2} = 4, b_{2} = - 6 and c_{2} = - 7$
$\frac{a_{1}}{a_{2}} = \frac{3}{4}, \frac{b_{1}}{b_{2}} = \frac{- 5}{- 6}$ and $\frac{c_{1}}{c_{2}} = \frac{- 6}{- 7}$
$\frac{a_{1}}{a_{2}} = \frac{3}{4}, \frac{b_{1}}{b_{2}} = \frac{5}{6}$ and $\frac{c_{1}}{c_{2}} = \frac{6}{7}$
As we know, the condition for the equations to have a unique solution is
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
Hence, the given system of pair of equations have a unique solution and the given equations are consistent.

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