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Negatively Triangular

Age 14 to 16 Challenge Level:


Patrick from Woodbridge school used a computer to check his answer:

"I used the Mac OS X application "Grapher" to help with this. I used it to plot the lines to check my answers. I did the intersections part by solving every equation against every other in a simultaneous equation, I got the answer of $6$ (I did each algebraically but it would take a lot of space to write up)"

You can view the four graphs using the Desmos graphing calculator here

Labelling the equations $1$ to $4$ in the obvious way, here are Patrick's co-ordinates of the six intersections:

$1$ and $2$: $(9,4)$
$1$ and $3$: $(1,7)$
$1$ and $4$: $(-47,25)$
$2$ and $3$: $(-1,-1)$
$2$ and $4$: $(-5,-3)$
$3$ and $4$: $(-2,-5)$

We can deduce that lines $2, 3$ and $4$ form the triangle.

Note that if we rearrange the equations into the standard form $y = mx + c$, it would be possible to sketch the four lines and estimate roughly where their intersections lie. It might then be sufficient to solve the problem without needing to explicitly calculate any co-ordinates.