Number of solutions for a pair of linear equations

Given the pair of linear equations:

1. $5x - 15y = 8$
2. $3x - 9y = \frac{24}{5}$

We can rewrite them in the standard form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

1. $5x - 15y - 8 = 0 \implies a_1 = 5, b_1 = -15, c_1 = -8$
2. $3x - 9y - \frac{24}{5} = 0 \implies a_2 = 3, b_2 = -9, c_2 = -\frac{24}{5}$

Now, let's compare the ratios of the coefficients: $\frac{a_1}{a_2} = \frac{5}{3}$

$\frac{b_1}{b_2} = \frac{-15}{-9} = \frac{15}{9} = \frac{5}{3}$

$\frac{c_1}{c_2} = \frac{-8}{-\frac{24}{5}} = \frac{8 \times 5}{24} = \frac{40}{24} = \frac{5}{3}$

Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{5}{3}$, the lines are coincident, which means they have infinitely many solutions.