Showing a point on a Cartesian plane also lies on a line

Question

I'm just doing some revision for an upcoming exam. Our lecturer has given us problems with solutions to help study for the exam. I've come across a problem which i don't understand.

Question:

Given the line , L1= (x,y) = (-1,-4) + (8,24)t

Show that the point D= (1,2) also lies on the line.

The solution provided from the lecturer is as follows:

(x,y) = (-1,-4) + (8,24)t
when t = 1/4
(x,y) = (-1,-4) + (2,6) = (1,2)
This proves (1,2) lies on L1.

I'm not able to understand how to get the solution:

t=1/4

Any help is appreciated.

Thanks

Answer 1

You want $(-1,-4)+(8,24)t=(1,2)$ for some value of $t$. This produces the system of equations: $$\begin{cases}-1+8t=1\\-4+24t=2\end{cases}\iff\begin{cases}8t=2\\24t=6\end{cases}$$ Can you solve this system for $t$?

Answer 2

A bit more geometrically: a point $p$ lies on the line $a+tu$ iff the vector joining $a$ and $p$, that is $p-a$, is parallel to $u$. In our case obviously $(2,6)\parallel(8,24)$.