From point to point
Can you combine vectors to get from one point to another?
Age
14 to 16
Challenge level
Problem
How can you get from $(10,66)$ to $(20,12)$
using only integer multiples of the vectors
$\begin{pmatrix}2\\-3\end{pmatrix}$ and $\begin{pmatrix}3\\2\end{pmatrix}$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
$x\begin{pmatrix}2\\-3\end{pmatrix} +y\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}20\\12\end{pmatrix} -\begin{pmatrix}10\\66\end{pmatrix}=\begin{pmatrix}10\\-54\end{pmatrix}$
$2x+3y=10$
$-3x+2y=-54$
$6x+9y=30$
$-6x+4y=-108$
$13y=-78$
$y=-6$
$x=14$
$\begin{pmatrix}10\\66\end{pmatrix}+14\begin{pmatrix}2\\-3\end{pmatrix} -6\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}20\\12\end{pmatrix}$