Simoultaneous equations

Apr 2010
34
1
Hey there,

Getting very stuck on solving these, ive tried using substitution but i dont know how to rearrange it...
can you help?

\(\displaystyle 137.9 = x + \frac{y}{\sqrt{0.05}}\)

and

\(\displaystyle 275.8 = x + \frac{y}{\sqrt{0.007}}\)
 
Nov 2009
927
260
Wellington
Hi,
to rearrange for \(\displaystyle y\) in the first equation, you can substract \(\displaystyle x\) then multiply by \(\displaystyle \sqrt{0.05}\). So you find :

\(\displaystyle 275.8 = x + \frac{\sqrt{0.05}(137.9 - x)}{\sqrt{0.007}}\)

So :

\(\displaystyle 275.8 = x + \frac{\sqrt{0.05}}{\sqrt{0.007}} (137.9 - x)\)

Setting \(\displaystyle a = \frac{\sqrt{0.05}}{\sqrt{0.007}}\) to make it a bit simpler, we are left with :

\(\displaystyle 275.8 = x + a (137.9 - x)\)

So :

\(\displaystyle 275.8 = x + 137.9a - ax\)

\(\displaystyle 275.8 - 137.9a = x - ax\)

\(\displaystyle 275.8 - 137.9a = x(1 - a)\)

\(\displaystyle \frac{275.8 - 137.9a}{1 - a} = x\)

Substituting back the value we chose for \(\displaystyle a\), we get :

\(\displaystyle x = \frac{275.8 - 137.9 \left ( \frac{\sqrt{0.05}}{\sqrt{0.007}} \right )}{1 - \frac{\sqrt{0.05}}{\sqrt{0.007}}} \approx 55.454\)

Finding the value of \(\displaystyle y\) is now straightforward :)

Does it make sense ? Remember not to get stopped by impressive square roots and stuff : as long as there are no \(\displaystyle x\) terms in them, they can be considered a constant (and thus substituted to some letter, \(\displaystyle a\) in my example)
 
Apr 2010
34
1
ah yes that helps thanks very much!
i see the way you got the unkown terms on one side, very nice ;)

Thanks again,

Max
 
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Dec 2007
3,184
558
Ottawa, Canada
Hey there,

Getting very stuck on solving these, ive tried using substitution but i dont know how to rearrange it...
can you help?

\(\displaystyle 137.9 = x + \frac{y}{\sqrt{0.05}}\)

and

\(\displaystyle 275.8 = x + \frac{y}{\sqrt{0.007}}\)
Since 275.8 is double 137.9, then you can go this simpler way:

x + y/sqrt(.007) = 2x + 2y/sqrt(.05)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Seeing "x" alone in both equations, the first thing I would think of is subtracting one equation from the other:
\(\displaystyle 275.8 = x + \frac{y}{\sqrt{0.007}}\)
\(\displaystyle 137.9 = x + \frac{y}{\sqrt{0.05}}\)

\(\displaystyle 137.9= \frac{y}{\sqrt{0.007}}- \frac{y}{\sqrt{0.05}}= y\left(\frac{1}{\sqrt{0.007}}- \frac{1}{\sqrt{0.05}}\right)\)

\(\displaystyle 137.9= y\left(\frac{\sqrt{0.05}- \sqrt{0.007}}{\sqrt{(0.05)(.007)}}\right)\)

\(\displaystyle y= \frac{137.9\sqrt{0.00035}}{\sqrt{0.05}- \sqrt{0.007}}\)