I have a system of equations that I don't know how to solve.

1) $x = a - y$ ;

2)$ y = b \times sin(90 -z)$;

3) $z = \dfrac{(x - c )^2 }{b^2 \times e^2}$

$a, b, c, d$ and $e$ are known. How can I solve for $x, y$ and $z$?

  • $\begingroup$ You could try substituting (1) into (3) and then (3) into (2). Then solve for $y$. Then use your result for $y$ to obtain the remaining unknowns. $\endgroup$
    – Pixel
    Jul 21 '14 at 8:37
  • $\begingroup$ pbs gave you the way. The problem will be solve the equation for $y$ and there will not be analytical solution. So, numerical methods, such as Newton, would probably be required. Also notice that $\sin(90-z)=\cos(z)$. Give me some numbers to play with. $\endgroup$ Jul 21 '14 at 8:43
  • $\begingroup$ @pbs But I don't know how to get y square out of sin. $\endgroup$
    – Mike Shaw
    Jul 21 '14 at 8:46
  • $\begingroup$ This is exactly the reason of my comment. $\endgroup$ Jul 21 '14 at 8:46
  • $\begingroup$ @ClaudeLeibovici For instance, a = 200, b = 50, c = 85 , no d , e = 2 $\endgroup$
    – Mike Shaw
    Jul 21 '14 at 8:53

As suggested by pbs, you eliminate $x$ for the first equation, replace it into the third equation to get $z$ and now you are left with $$y=\cos \left(\frac{(a-c-y)^2}{b^2 e^2}\right)$$ which not the most pleasant I know.

Let me use your numbers $a=200,b=50,c=85,e=2$. So, the equation is $$y=\cos \left(\frac{(115-y)^2}{10000}\right)$$ and so you want to solve for $y$ $$f(y)=y-\cos \left(\frac{(115-y)^2}{10000}\right)=0$$ What you can first notice is that $$f(0)=-\cos \left(\frac{529}{400}\right) \simeq -0.245753$$ Looking at the plot of f(y) as a function of $y$ reveals an almost linear behavior.

For solving nonlinear equations such as the present one, Newton method is one of the simplest tools. Starting from a "reasonable" guess $y_0$, the procedure will update it according to $$y_{n+1}=y_n-\frac{f(y_n)}{f'(y_n)}$$ For the equation we look for $$f'(y)=\frac{(y-115) \sin \left(\frac{(115-y)^2}{10000}\right)}{5000}+1$$ So, let us apply Newton starting at $y=0$; this will generate the following iterates :$0.251357$, $0.251346$ which is the solution for six significant figures.

So, now, using $y=0.251346$, going backward, we find $x=199.749$ and $z=1.31673$.

What you also can do is to define $$y_{n+1}=\cos \left(\frac{(115-y_n)^2}{10000}\right)$$ and start with $y_0=0$. In this case, the following iterates will be obtained : $0.245753$, $0.251222$, $0.251344$, $0.251346$.

  • $\begingroup$ My question is more like how to express x y z with a b c and e. Btw, I'm so sorry for not telling you z is in degree so is 90. But I don't know how to edit the question to add degree. $\endgroup$
    – Mike Shaw
    Jul 21 '14 at 9:24
  • $\begingroup$ What you are being told, Mike, is that you can't express $x,y,z$ in terms of $a,b,c,e$. All you can do is find a numerical solution when given particular values of $a,b,c,e$. $\endgroup$ Jul 21 '14 at 10:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.