PROJECT: Systems of Linear Equations
Objective:
Solve a system of Linear equations using Gauss-Jordan Elimination
The Terraforming Problem
A Terraforming expedition is attempting to convert the
alien soil of an uninhabited planet into its Earth
counterpart. It has 4 different mixtures of soil additives
available.
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A container of Mixture A contains 30 kilotons of
phosphoric acid, 50 kilotons of nitrogen, and 30
kilotons of potash.
A container of Mixture B contains 30 kilotons of
phosphoric acid, 75 kilotons of nitrogen, and 40
kilotons of potash.
A container of Mixture C contains 60 kilotons of
phosphoric acid, 25 kilotons of nitrogen, and 50
kilotons of potash.
A container of Mixture D contains 30 kilotons of
phosphoric acid, 25 kilotons of nitrogen, and 20
kilotons of potash.
Soil tests indicate that this particular area of alien soil needs 1500 kilotons of phosphoric acid, 2050
kilotons of nitrogen, and 1400 kilotons of potash to make it similar to Earth's soil. Then food crops
can be planted. How many containers of each mixture should the terraformers use to supply the
alien soil with the necessary nutrients to convert it to Earth standards? (Note: Fractions of a
container cannot be used.)
Project:
Use Gauss-Jordan Elimination (rref) to solve the "terraforming" problem satisfying the following
criteria:
A. Translate the problem into a linear system. Show the system of linear equations
B. Convert the linear system into an augmented matrix.
C. By hand, perform the Gauss-Jordan Elimination on the first column. Get a 1 in
position (1,1) and zeroes underneath it. Show the row commands and your work
for each step.
D. Going back to the original augmented matrix, use your calculator (rref) to solve the
linear system.
E. List
1. The general form of the solution showing the free (arbitrary) variable.
2. The limits of the arbitrary (free) variable.
Note: What values of the arbitrary variable will make the other variables
positive? You cannot have a negative number of containers.
3. Using only whole numbers for the arbitrary (free) variable, list all specific
solutions to the problem.