Question

Find whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident :
$\frac{3}{2}x+\frac{5}{3}y=7$, $\frac{3}{2}x+\frac{2}{3}y=6$

Answer 1

The 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 $a_1=\frac{3}{2}$,$b_1=\frac{5}{3}$ and $c_1=7$

And $a_2=\frac{3}{2}$,$b_2=\frac{2}{3}$ and $c_2=6$

We note that $\frac{a_1}{a_2}=\frac{\frac{3}{2}}{\frac{3}{2}}=1$

$\frac{b_1}{b_2}=\frac{\frac{5}{3}}{\frac{2}{3}}=\frac{5}{2}$

and $\frac{c_1}{c_2}=\frac{7}{6}$

Hence $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

$\therefore$ the given lines are intersecting.