2.4 Applications of Systems of Linear Equations

Uses everything 2.2 RREF 2.1 REF 2.2 Rank 2.2 Homogeneous Equations

Water Flow example

• At each node, water flow in = water flow out

  • 
  • what are the possible solutions for $x_1,x_2,x_3,x_4$ below?
  • if $x_1=10$ what are the other flows?

• setup

$$\begin{array}{r}A:10+20=x_1+x_4\\ B:x_1=10+10+x_2\\ C:x_3+x_4=35+25\\ D:25+25+x_2=x_3\end{array}⟹\left[\begin{array}{ccccc}1& 0& 0& 1& |30\\ 0& -1& 0& -1& |-10\\ 0& 0& 1& 1& |60\\ 0& 1& -1& 0& |-50\end{array}\right]$$

• Doing RREF we get the vector

$$RREF\left[\begin{array}{ccccc}1& 0& 0& 1& |30\\ 0& 1& 0& 1& |10\\ 0& 0& 1& 1& |60\\ 0& 0& 0& 0& |0\end{array}\right]$$

- keeping in mind the coloumns are $x_{1},x_{2},x_{3},x_{4}$
	- we can see that $x_{4}$ is a free variable because its colloumn has no leading 1s
	- Let 

$$\begin{array}{r}{x}_4=t\\ x_1=30-t\\ x_2=10-t\\ x_3=60-t\end{array}⟹\left[\begin{array}{c}x1\\ x2\\ x3\\ x4\end{array}\right]=\left[\begin{array}{c}30\\ 10\\ 60\\ 0\end{array}\right]+t\left[\begin{array}{c}-1\\ -1\\ -1\\ 1\end{array}\right](t\in \mathbb{R})$$

• if $x_1=10$ the

$$\begin{array}{r}{x}_1=10\\ x_2=10-20\\ x_3=60-20\\ x_4=t=20\end{array}$$

- we can see that the internal flow rates d=>b is reversed and the resulting numbers from the set above shows the internal flow

Partial Fraction (WHAT)

• Find values of A and B below that satisfy the partial fraction equation $=$

• (using the values at x=0)

  • we can make it so that all rows show the powers of x and using turn it into a matrix on if any of the varibles are tthere on that specific power

$$\left[\begin{array}{cccc}1& 1& 0& |0\\ 0& 2& 1& |0\\ 1& 0& 2& |1\end{array}\right]⟹RREF$$

Polynomial interpolation 2.1

• Find the quadritic function $y=a{x}^2+bx+c$ that passes through the points $(-1.1),(0,1),(1,2)$

  • we can just make a matrix with all the valuesand what it means for A, B, C

$$\begin{array}{r}a-b+c=1\\ 0a+0b+c=1\\ a+b+c=2\end{array}⟹\left[\begin{array}{cccc}1& -1& 1& |1\\ 0& 0& 1& |1\\ 1& 1& 1& |2\end{array}\right]$$

	- then with that matrix we can find the resulting REF/RREF that gives us that 

$$\begin{array}{r}a=\frac{1}{2}\\ b=\frac{1}{2}\\ c=1\\ ⟹y=\frac{1}{2}{x}^2+\frac{1}{2}x+1\end{array}$$

Balancing chemical equations

• Find coefficeients $x_1,x_2,x_3,x_4$

$$\begin{array}{r}{x}_1{C}_4{H}_6+x_2{O}_2\to x_{3}C{O}_2+x_4{H}_{2}O\\ C:6x_1=x_3\\ H:6x_1=2x_4\\ O:2x_2=2x_3+x_4\end{array}$$

- This creates a [[2.2 Homogeneous Equations]] 

$$\left[\begin{array}{ccccc}6& 0& -1& 0& |0\\ 6& 0& 0& -2& |0\\ 0& 2& -2& -1& |0\end{array}\right]\to REF\left[\begin{array}{ccccc}1& 0& -1/6& 0& |0\\ 0& 1& -1& -1/2& |0\\ 0& 0& 1& -2& |0\end{array}\right]$$