Home | | Latest | About | Random
# Linear system of equations problems.
Let us practice some of the things we learned today: Linear system of equations, how to solve them using elementary row operations, and echelon forms.
![[---images/---assets/---icons/question-icon.svg]] For each of the following linear system of equations, (a) find all possible solutions (b) does it have unique solution, infinitely many solutions, or no solutions?
$$
(1)\quad\left\{\begin{array}{}
x & - & 2y & + & 3z & = & 9 \\
-x &+ & 3y & & & = & -4 \\
2x & - & 5y & + & 5z & = & 17
\end{array}\right.
$$$$
(2)\quad\left\{
\begin{array}{}
x & - & y & + & 3z & = & 4 \\
2x & + & y & + & 3z & = & 1
\end{array}
\right.
$$$$
(3)\quad\left\{
\begin{array}{}
4x & + & 12y & - & 7z & - &20w & = & 22 \\
3x & + & 9y & - & 5z & - & 28w & = & 30
\end{array}
\right.
$$$$
(4)\quad\left\{
\begin{array}{}
x & + & 2y & = & 0 \\
x & + & y & = & 6 \\
3x & - & 2y & = & 8
\end{array}
\right.
$$
![[---images/---assets/---icons/question-icon.svg]] Is it necessarily true that whenever we have more equations than variables, then we must have no solution? Justify or give a counterexample.
![[---images/---assets/---icons/question-icon.svg]] **A zoo of animals.** Consider the following information:
A crab has 8 legs, 2 arms, no wings, and no tail.
A duck has 2 legs, no arms, 2 wings, and one tail.
A dragon has 2 legs, 2 arms, 2 wings, and one tail.
A mermaid has no legs, 2 arms, no wings, and one tail.
In a collection of some of these four creatures, there are 174 legs, 46 arms, 78 wings , and 45 tails in total. How many of each creature are there? (And how many ducks be quacking?)
(Set up a system of linear equations, what are your unknown variables?)
![[summer program 2023/---files/Pasted image 20230905110748.png]]
![[---images/---assets/---icons/question-icon.svg]] **Balancing chemical equations.** Quite often a chemical equation can be balanced by ad hoc methods and "just looking at it". But what you are doing is actually solving a system of linear equation!
Take a look at this simple example: A combustion reaction of methane $CH_4$ with oxygen $O_2$, producing carbon dioxide $CO_2$ and water $H_2O$ : $$
CH_{4} + O_{2} \to CO_{2} + H_{2}O
$$
Now above is not yet balanced. We need to find coefficients $x,y,z,w$ such that $$
xCH_{4} + yO_{2} \to zCO_{2} + wH_{2}O
$$
is balanced. To do this, set up a linear equation for each element (so you should have four), as each element needs to be balanced on both sides. For example looking at carbon, we need to have $x=z$ (why?).
Once you have set it up as a linear system equation as the usual way, say aligning all the variables in an augmented matrix, solve that system.
If you do it correctly, you would get infinitely many solutions (this is because any scaling of a balanced equation is alsoop balanced!), set the free variable to a suitable number to get a **least positive integer solution**, since this is what we typically look for when balancing a chemical equation.
If you know how to balance a typical hydrocarbon combustion reaction, see if it matches with what you already know. Note, this method is not necessarily the fastest, but can be reliable.
![[---images/---assets/---icons/question-icon.svg]] Balance the following chemical reactions: $$
Al_{2}O_{3} + HF \to AlF_{3}+ H_{2}O
$$$$
KOH+H_{3}PO_{4}\to K_{3}PO_{4}+H_{2}O
$$$$
K_{2}MnF_{6} + SbF_{5} \to KSbF_{6} + MnF_{3} + F_{2}
$$$$
KNO_{3}+C_{12}H_{22}O_{11}\to N_{2}+CO_{2}+H_{2}O + K_{2}CO_{3}
$$
![[---images/---assets/---icons/question-icon.svg]] **Polynomial fitting.** A third degree real polynomial $p(z)$ in variable $z$ has a generic form$$p(z) = a z^3 + b z^2 + c z + d,$$for some real numbers $a,b,c,d$. Suppose such a third degree polynomial goes through the points $(z,p(z))$ at $(0,1), (3,1),(-1,0),(1,2)$, determine this third degree polynomial. That is, find these unknown coefficients $a,b,c,d$.
(Hint plug in the points to see if it reveals a system of linear equation for $a,b,c,d$.)
![[---images/---assets/---icons/question-icon.svg]] Consider all $2\times 2$ binary matrices whose entries are either 1 or 0. That is, matrices of the form$$
\begin{bmatrix}a & b\\
c & d
\end{bmatrix}
$$where $a,b,c,d$ is either 0 or 1.
(a) How many of these $2\times 2$ binary matrices are there in total?
(b) Among these, how many of these are in EF?