# How to determine the number of solutions in a linear system without solving: 2x - 3y = 4 and 6y = 4x + 15?

Nov 11, 2015

No solution

#### Explanation:

First, let's move either $x$ or $y$ (or both, whichever you find convenient) such that the position of the variables for both equations are the same.

$2 x - 3 y = 4$

$6 y = 4 x + 15$

$\implies - 4 x + 6 y = 15$

If we check the ratio between the coefficients of $x$ and $y$, we can immediately see that both equations have the same slope. We can conclude that the lines are parallel

$2 x - 3 y = 4$

$\implies m = \frac{2}{3}$

$- 4 x + 6 y = 15$

$\implies m = \frac{4}{6} = \frac{2}{3}$

What remains to be checked is whether the lines are coincidental or not. If the lines are coincidental, there are infinitely many solutions. Otherwise, there is no solution. If we check out the constant for both equations, we can see that they are not the same

$2 x - 3 y = 4$

$- 4 x + 6 y = 15$

$- 2 \left(2 + 3 y\right) = - 2 \left(- \frac{15}{2}\right)$

We can now conclude that the lines are not coincidental.
Hence, there is no solution.

Yes, we did a little bit of "solving". But once you get used to it, you'll be able to know without solving.