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Discovery guide 3.1 Discovery guide
In this set of discovery activities, we look at some places where linear systems naturally arise.
Discovery 3.1 .
Use the Law of Conservation (in this case, flow in equals flow out at each point of intersection) in the flow network below to set up a system of equations to determine the internal flow rates. (Do not solve your system.)
Discovery 3.2 .
Set up a system of equations to balance the chemical equation:
\begin{equation*}
a\mathrm{NH_3} + b\mathrm{O_2} \rightarrow c\mathrm{NO} + d\mathrm{H_2O}.
\end{equation*}
Do not solve your system.
No shortcuts. You might have learned some procedure for balancing a chemical equation in high school. We are not interested in that procedure. We would like to see how attempting to balance a chemical equation has a linear system at its root.
Discovery 3.3 .
Any two (distinct) points in the Cartesian plane determine a unique line. Set up a system of equations that would let you solve for the slope and \(y\) -intercept of the line \(y=mx+b\) that passes through the points \((-3,4)\) and \((2,-1)\) (but do not solve the system). Write down the augmented matrix for your system.
Discovery 3.4 .
Any three (distinct, non-collinear) points in the Cartesian plane determine a unique parabola. Set up a system of equations that would let you solve for the coefficients \(a,b,c\) of the parabola \(y=ax^2+bx+c\) that passes through the points \((-1,-4)\text{,}\) \((1,0)\text{,}\) and \((2,5)\) (but do not solve the system). Write down the augmented matrix for your system.
Discovery 3.5 .
Based on the previous two activities and their answers, how many points are necessary to determine a unique degree \(n\) polynomial \(y=a_n x^n + a_{n-1} x^{n-1} + \dotsb + a_1 x + a_0\text{?}\) If you knew such points
\begin{equation*}
(x_1,y_1),(x_2,y_2),(x_3,y_3),\dotsc,
\end{equation*}
and you used these points to create a linear system to determine the coefficients of the polynomial, what would be the pattern in the rows of the resulting augmented matrix for the system?
