Sorry

little mistake in the end

must be by = xc ( and this is 2017)

]]>Since

ax = bz (=20) ===> b = ax / z

and

ay = cz (=17) ===> y = cz / a

well we are looking for by:

by = (axcz) / (az) ===> by = xz (and this is 2017)

]]>You can “solve” it by hand in the sense that the system is underdetermined, hence one of the variables is free. Pick your independent variable, then solve everything else in terms of that variable.

By the way, by hand, I get:

b*y

= (20/z)*y (since b*z = 20)

= (20/17)*c*y (since c*z = 17)

= (20/17)*(2017/x)*y (since c*x = 2017)

= (2017/17)*a*y (since a*x = 20)

= 2017 (since a*y = 17).

Maple, by the way, does a more complete job of describing all possible solutions, and returns

{a = 17/y, b = 2017/y, c = 34289/(20*y), x = (20/17)*y, y = y, z = (20/2017)*y}

]]>You can’t really “solve” the system by hand, since there are six unknowns but only five equations. I didn’t intend for you to use Maple, but I’m curious… what values did it return for each variable?

I had an algebraic trick in mind to find the solution, though working with my sons last night, they were far more keen on trying to find values for a, b, c, x, y, and z and then using those values to determine b*y.

]]>One could solve this system by hand, but that seems rather tedious, and therefore can’t possibly be the intended method of attack.

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