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Solving Systems of Equations with Matrices II
When given a system of linear equations, you can find their point of intersection via matrices. When it would take hours for a person to solve a many-variable system with substitution, it takes, at most, a couple of minutes with matrices. Matrices should not be your default method of solving systems, since other methods might be faster than typing the matrices into your calculator.
Use matrices to find (x,y) if

  1. Form into an augmented matrix.



  2. Use row operations to get the augmented matrix into reduce row echelon form.



  3. By putting it back into equation form, the answer is revealed.



  4. Show answers as an ordered pair.

Matrices can also be used to find solutions to word problems. Set up a system of equations that correspond to the data found within the word problem and then solve the system using matrices as discussed above.

Bill, a very observant yet clumsy man with a wad of cash in his hands, was just leaving from getting his wash done at the local laundry mat when he bumped into another person holding his own wad of cash. Currency flew everywhere. The other man began to get very irate with Bill; he wanted his money. Bill thought back and remembered entering with 150 dollars in coins and bills. He thought back some more and recalled spending all $9 worth of coins he had been carrying and not carrying any bills larger than a fifty. He recalled counting 11 bills, having a total of 5 Washington DC monuments on the back sides of the bills, 3 pictures of presidents with beards, and 92 letters worth of presidential last names. Even with all of that memory, he failed to remember what bills he had. Find the correct denomination of bills Bill was carrying. You can use this link to view a collection of US banknotes.

    1. Write a system of linear equations, where a is the number of one dollar bills, b is the number of five dollar bills, c is the number of ten dollar bills, d is the number of twenty dollar bills and e is the number of fifty dollar bills.

      The first equation represents the number of bills in his wallet.
      The second equation represents the monetary value of the bills.
      The third equation represents the number of bills that had monuments on their backs.
      The fourth equation represents the number of bills that had pictures of presidents with beards.
      The final equation represents the total number of letters in the presidential names.

    2. Form the system into an augmented matrix; remember to fill in the missing terms with zeros.



    3. Use or the calculator to get the matrix into reduced row echelon form.


      1. Press .




      2. Press  and  to access the matrix menu.




      3. Arrow to “EDIT” tab.




      4. Highlight [A] and press .




      5. Type , , to set the proper dimensions of the new matrix.




      6. Override the existing values by highlighting an entry and typing. Type the augmented matrix from step 2 into the calculator.




      7. Press    to return to the home screen.




      8. Press    to access the matrix menu.




      9. Arrow over to the “MATH” tab.




      10. Arrow down to B: rref (.




      11. Press .




      12. Press    to access the matrix menu.




      13. Press .




      14. Press  for it to calculate the reduced row echelon form of the matrix.




    4. Put your results back in equation form.

      This means that Bill was carrying 6 ones, 1 five, 1 ten, 1 twenty, and 2 fifties.
 

Example Group #1
Bill now has a group of pennies, nickels, dimes, and quarters totaling $1.19 in value. There are a total of 16 coins, 10 with a picture of some Washington D.C. monument and 7 with a picture of a man looking to our left.
Example How many pennies, nickels, dimes, and quarters does Bill have?
What is your answer?
 

Example Group #2
Soup Herman was flying around the sky above downtown Reading, Pennsylvania, when he heard a cry for help from pedestrians on the ground. He used his trademark Get-It-Done-Now-Before-Someone-Gets-Hurt Sense and mentally visualized an impact-activated incendiary device headed for a crowded sidewalk. Within an instant, Soup Herman was under the falling explosive keeping it from hitting the ground. After continuing to push up on the bomb, he was able to launch it into outer space were it would safely detonate.



The formula for finding an object’s position during acceleration is:

The formula for finding an object’s velocity during acceleration is:

In these equations
  • a represents acceleration,
  • s represents position,
  • t represents time, and
  • v represents velocity.
Example
Determine Soup Herman's velocity one second after he makes contact with the explosive.
What is your answer?
 

Example Group #3
The following table contains information about three metals.

Symbolg/molMelting point (K)Price/mol
X 10.23 2140 $42.00
Y 8.25 1980 $15.25
Z 1.43 925 $36.70


Example A company wants to make an alloy from the three metals above. They want it to have a weight of 5 g/mol and a melting point of 1200K. They only want to pay $15.00 per mole of the new alloy. How much of each metal should be mixed together for a metal alloy that fits the requests of the company?
What is your answer?
 



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