How to find intersection of two lines in parametric form?

Question

I want to find the intersection of $ \mathrm L(x,y) = 3x + 4y - 8 = 0$ and $\mathrm M(x,y) = 2x - 5y + 3 = 0$.

Let parametric form of $\rm L$ is $\mathbf Q(t) = \vec a + t(-4, 3)$ and $\mathbf R(t) = \vec a + t(5, 2)$ be the parametric form of $\mathrm M$. $\vec a$ is the intersection of both lines.

If I put $\mathbf Q(t_0) = \mathbf R(t_0)$ then I get $t_0 = 0$. How do I find $\vec a$ from this ? If I put $t_0$ back in either of the $\bf Q$ or $\bf R$ I will get $\mathbf {R}(t_0) = \mathbf Q(t_0) = \vec a$, which is not very helpful.

I know the intersection is $\bigg(\dfrac {84}{69}, \dfrac {25}{23}\bigg)$.

Answer 1

You must solve the system of equations: $$\begin{cases}-4t=1+5s \\ 3+2t=1+2s\end{cases}$$