0
$\begingroup$

Solve the system \begin{cases} \phantom6x_1 + \phantom7x_2 + 3x_3 = 5,\\[0.3em] 6x_1 + 7x_2 - 3x_3 = 5; \end{cases}

I need the solution in the form of \begin{cases} x_1=\color{gray}{\text{blank}}+\color{gray}{\text{blank}}\times s;\\[0.3em] x_2=\color{gray}{\text{blank}}+\color{gray}{\text{blank}}\times s;\\[0.3em] x_3=\color{gray}{\text{blank}}+\color{gray}{\text{blank}}\times s. \end{cases}

Very similar to a question I have previously asked with two variables. My attempt is to set it up in a matrix and eliminate:

\begin{align} &\left[\begin{array}{ccc|c} 1 & 1 & \phantom-3 & 5 \\ 6 & 7 & -3 & 5 \\ \end{array}\right] % \begin{array}{}\\+\text{row}_1\end{array}\\ \iff&\left[\begin{array}{ccc|c} 1 & 1 & 3 & \phantom15 \\ 7 & 8 & 0 & 10 \\ \end{array}\right] % \begin{array}{}\\-7\text{row}_1\end{array}\\ \iff&\left[\begin{array}{ccc|c} 1 & 1 & \phantom{-2}3 & \phantom{-2}5 \\ 0 & 1 & -21 & -25 \\ \end{array}\right] % \begin{array}{}-\text{row}_2\\{}\end{array}\\ \iff&\left[\begin{array}{ccc|c} 1 & 0 & \phantom-24 & \phantom-30 \\ 0 & 1 & -21 & -25 \\ \end{array}\right] % \begin{array}{}\\.\end{array} \end{align}

Would I then end up with the following?

\begin{cases} x_1 = \phantom- 30 - 24s,\\[0.3em] x_2 = -25 + 21s,\\[0.3em] x_3 = \phantom{-2}0 + \phantom20s. \end{cases}

$\endgroup$
1
$\begingroup$

You should set $x_3 = s$.

Think of $x_3$ as a free variable. You started with two equations and three unknowns; this means you have (at least) one free variable. It didn't have to be $x_3$, but it might as well be.

When you write $x_3 = 0+0s$, you are saying $x_3$ has to be zero for the system to hold; while in fact $x_3$ can take on any value, and the system will be satisfied as long as $x_1$ and $x_2$ behave correctly according to $x_3$ (or $s$).

$\endgroup$
  • $\begingroup$ I follow the setting x3=s, but not sure how that would then apply to answering in the format that is required. $\endgroup$ – David Scidmore Feb 19 '16 at 1:02
  • $\begingroup$ $x_3 = s$ is the same as $x_3 = 0+1\cdot s$, which is the correct format @davidscidmore $\endgroup$ – GaussTheBauss Feb 19 '16 at 1:05
  • $\begingroup$ Thank you very much, I just wasn't sure if that was what it needed to be. You also helped me with about 5 different problems since my instructor seems to not care. @gaussthebauss. $\endgroup$ – David Scidmore Feb 19 '16 at 1:10
  • $\begingroup$ @davidscidmore you are welcome $\endgroup$ – GaussTheBauss Feb 19 '16 at 1:11

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.