Augmented Matrices and The Gauss-Jordan Method
(Teacher Notes)
Use the information below to set up a system of equations and then solve the system using the elimination method.
An investor has a total of 45 one-ounce ingots, made of either gold or silver, worth $7636.50. The value of a gold ingot is $280.00, and the value of a silver ingot is $4.25. Find g, the number of gold ingots, and s, the number of silver ingots?
(Released TAKS – July 2006)
System of EquationsAugmented Matrices
Answer:
1st : Find s by eliminating g.
2nd : Find g by eliminating s in R1.
The process used above is called The Gauss-Jordan Elimination Method. This is a systematic way to solve system of equations and is especially helpful when solving very large systems of equations. The first step in using the Gauss-Jordan Elimination Method is to use the system of equations to set up an augmented matrix (a coefficient matrix next to a constant matrix) . For the problems below, set up a system of equations and the corresponding augmented matrix.
Problem #1: An isosceles triangle has legs that are eachx inches long and a base that is y inches long. The perimeter of this triangle is 38 inches. The base is 8 inches shorter than the length ofa leg. Find the length of each of the three sides. (Released TAK Fall 2005)
Answer: Let x = length of the leg (in inches)
y = length of the base (in inches)
Problem #2: A toy store set up some bargain tables in the shopping mall. Three children, Ali, Bob, and Cindy, bought some items. On the basis of the following information, determine how many items each child purchased.
- The total number of items purchased was 5.
- Ali paid $1 for each item he purchased, Bob paid $2 each and Cindy paid $3 each. The total spend by all three children was $10.
- The toy store gave balloons when items were purchased. Ali got 2 balloons for each item he purchased; Bob and Cindy got 1 balloon for each item. The children received a total of 6 balloons.
Answer: Let a = number of items purchased by Ali
b = number of items purchased by Bob
c = number of items purchased by Cindy
Problem #3: A theater has a seating capacity of 900 and charges $2 for children, $3 for students, and $4 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $2800. How many children attended the show?
Answer: Let a = number of adults that attended the show
s = number of students that attended the show
c = number of children that attended the show
The goal in Gauss-Jordan Elimination is to use row operations (interchange two rows, multiply a row by a nonzero constant, add two rows, or add a multiple of one row to another row) tochange the augmented matrix to row reduced echelon form (rref) which looks like the following matrix:
Notice the right hand side is the identity matrix while the left hand side can be any numbers. Why do you think we want to convert the augmented matrix to the ‘identity-like” matrix above?
To get a matrix in row reduce echelon form follow the method suggested below.
First:
Second:
Third:
Use the augmented matrices set up above and perform row operations to get each of the augmented matrices in row reduced form.
Problem #1: An isosceles triangle has legs that are eachx inches long and a base that is y inches long. The perimeter of this triangle is 38 inches. The base is 8 inches shorter than the length ofa leg. Find the length of each of the three sides. (Released TAK Fall 2005)
The length of the legs is 46/3 inches and the length of the base is 22/3 inches.
Problem #2: A toy store set up some bargain tables in the shopping mall. Three children, Ali, Bob, and Cindy, bought some items. On the basis of the following information, determine how many items each child purchased.
- The total number of items purchased was 5.
- Ali paid $1 for each item he purchased, Bob paid $2 each and Cindy paid $3 each. The total spend by all three children was $10.
- The toy store gave balloons when items were purchased. Ali got 2 balloons for each item he purchased; Bob and Cindy got 1 balloon for each item. The children received a total of 6 balloons.
Ali bought one item, Bob bought three items, and Cindy bought one item.
Problem #3: A theater has a seating capacity of 900 and charges $2 for children, $3 for students, and $4 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $2800. How many children attended the show?
Augmented Matrices and The Gauss-Jordan Method
(Student Notes)
Use the information below to set up a system of equations and then solve the system using the elimination method.
An investor has a total of 45 one-ounce ingots, made of either gold or silver, worth $7636.50. The value of a gold ingot is $280.00, and the value of a silver ingot is $4.25. Find g, the number of gold ingots, and s, the number of silver ingots?
(Released TAKS – July 2006)
System of EquationsAugmented Matrices
1st : Find s by eliminating g.
2nd : Find g by eliminating s in R1.
The process used above is called The Gauss-Jordan Elimination Method. This is a systematic way to solve system of equations and is especially helpful when solving very large systems of equations. The first step in using the Gauss-Jordan Elimination Method is to use the system of equations to set up an augmented matrix (a coefficient matrix next to a constant matrix) . For the problems below, set up a system of equations and the corresponding augmented matrix.
Problem #1: An isosceles triangle has legs that are eachx inches long and a base that is y inches long. The perimeter of this triangle is 38 inches. The base is 8 inches shorter than the length ofa leg. Find the length of each of the three sides. (Released TAK Fall 2005)
Problem #2: A toy store set up some bargain tables in the shopping mall. Three children, Ali, Bob, and Cindy, bought some items. On the basis of the following information, determine how many items each child purchased.
- The total number of items purchased was 5.
- Ali paid $1 for each item he purchased, Bob paid $2 each and Cindy paid $3 each. The total spend by all three children was $10.
- The toy store gave balloons when items were purchased. Ali got 2 balloons for each item he purchased; Bob and Cindy got 1 balloon for each item. The children received a total of 6 balloons.
Problem #3: A theater has a seating capacity of 900 and charges $2 for children, $3 for students, and $4 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $2800. How many children attended the show?
The goal in Gauss-Jordan Elimination is to use row operations (interchange two rows, multiply a row by a nonzero constant, add two rows, or add a multiple of one row to another row) to change the augmented matrix to row reduced echelon form (rref) which looks like the following matrix:
Notice the right hand side is the identity matrix while the left hand side can be any numbers. Why do you think we want to convert the augmented matrix to the ‘identity-like” matrix above?
To get a matrix in row reduce echelon form follow the method suggested below.
First:
Second:
Third:
Use the augmented matrices set up above and perform row operations to get each of the augmented matrices in row reduced form.
Problem #1: An isosceles triangle has legs that are eachx inches long and a base that is y inches long. The perimeter of this triangle is 38 inches. The base is 8 inches shorter than the length ofa leg. Find the length of each of the three sides. (Released TAK Fall 2005)
Problem #2: A toy store set up some bargain tables in the shopping mall. Three children, Ali, Bob, and Cindy, bought some items. On the basis of the following information, determine how many items each child purchased.
- The total number of items purchased was 5.
- Ali paid $1 for each item he purchased, Bob paid $2 each and Cindy paid $3 each. The total spend by all three children was $10.
- The toy store gave balloons when items were purchased. Ali got 2 balloons for each item he purchased; Bob and Cindy got 1 balloon for each item. The children received a total of 6 balloons.
Problem #3: A theater has a seating capacity of 900 and charges $2 for children, $3 for students, and $4 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $2800. How many children attended the show?