0
$\begingroup$

The question goes as the following $$x-3y+6z=21$$ $$3x+2y-5z=-30$$ $$2x-5y+2z=-6$$

for this one I am really confused and don't even know how to solve it.

I tried to do the elimination method but i came with fractions.

I woud show the process but I'm currently using my phone.

$\endgroup$
  • $\begingroup$ do you know the Gauss-method? $\endgroup$ – Dr. Sonnhard Graubner Sep 15 '16 at 14:46
  • 1
    $\begingroup$ What's the problem with fractions? $\endgroup$ – barak manos Sep 15 '16 at 14:47
  • $\begingroup$ No I don't know what that means $\endgroup$ – MATH ASKER Sep 15 '16 at 14:47
2
$\begingroup$

$$\left\{\begin{array}{lcr}(A)\;\phantom{1}x-3y+6z&=&21\\ (B)\;3x+2y-5z&=&-30\\ (C)\;2x-5y+2z&=&-6\end{array}\right.$$ is equivalent (through $(B)\mapsto(B-3A)$ and $(C)\mapsto (C-2A)$) to$$\left\{\begin{array}{lcr}(A)\;\phantom{1}x-3y+6z&=&21\\ (B)\;\phantom{3x+}11y-23z&=&-93\\ (C)\;\phantom{2x-\,\,\,}y-10z&=&-48\end{array}\right.$$ that is equivalent (through $(B)\mapsto(B-11C)$) to $$\left\{\begin{array}{lcr}(A)\;\phantom{1}x-3y+6z&=&21\\ (B)\;\phantom{3x+11y-}87z&=&435\\ (C)\;\phantom{2x-\,\,\,}y-10z&=&-48\end{array}\right.$$ Now use $(B)$ to find the value of $z$, plug it into $(C)$ to find the value of $y$, plug them both into $(A)$ to find the value of $x$. This is known as the Gaussian elimination method. A good alternative for $3\times 3$ systems is Cramer's rule. They both lead to: $$ (x,y,z)=\color{red}{(-3,2,5)} $$

$\endgroup$
  • 1
    $\begingroup$ Thank you very much, This helped me a lot $\endgroup$ – MATH ASKER Sep 15 '16 at 21:38
0
$\begingroup$

multiplying the first equation by -3 and adding to the seconde one we get $$11y-23z=-93$$ (I) multiplying the first by -2 and adding to the third we get $$y-10z=-48$$ (II) can you proceed? multiplying (II) by -11 and adding this to (I) we obtain $$87z=435$$ thus $$z=5$$

$\endgroup$
0
$\begingroup$

You can row reduce the matrix with no fractions to

$$\left[\begin{array}{ccc|c} 1&0&0&-3\\ 0&1&0&2\\ 0&0&1&5\\ \end{array}\right]$$

giving $x=-3, y=2,z=5$

$\endgroup$

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.