Math 141-copyright Joe Kahlig, 10B Page 1Section 2.2: Systems of Linear Equations: Unique SolutionsSection 2.3: Systems of Linear Equations: Underdetermined and Overdetermined SystemsSolving System of Equations.Definition: An augmented matrix is a condens ed method of representing a system of equations.Example: Represent the system of equations as an augmented matrix.3x + 2y = 7x + 4y = 10Example: Give the system of equations rep resented by the augmented matrix. (Variables are listedin the first r ow.)x y z1 -4 2 103 4 220Row Operations are us ed to manipulate an augmented matrix into a form (usually row reducedform) wh ere the solution can easily be discerned. The three row operations are:1) Swap Rows2) Multiply a row by a non-zero number3) Add a multiple of one row to another row.Row Reduced Form (reduced row echelon form)1. The first non-zero number in a row is a 1(called a leading one).2. The leading one is the only non-zero number in a column.3. The leading ones are in a diagonal like fashion from the upper left to the lower right.Example: Which of these matrices are in row reduced form?A)1 0 030 1 080 0 12B)1 0 350 1 060 0 27C)"1 3 070 0 16#D)1 0 030 1 0 70 0 010Math 141-copyright Joe Kahlig, 10B Page 2Example: Solve th e system of equations.3x + 2y = 7x + 4y = 10Example: Create a leading one in the row one column one position.3x + y + 2z = 114x + 9y + z = 252x − y + 3z = 9Math 141-copyright Joe Kahlig, 10B Page 3Example: Solve th ese system of equations.A) 3x + y − 9 = 0x − y + z − 4 = 03x + z − 11 = 04x − y + 2z = 15B) x + y − 3z = 02x − 3y + z = 14x − y − 5z = 1C) x + 3y − z − 3w = 72x + 4y − 2w = 10Math 141-copyright Joe Kahlig, 10B Page 4Example: The fi gure shows the flow of traffic durin g rush hour on a typical weekday. The arrowsindicate the direction of traffic flow on each one–way road, and the average number of vehicles enteringand leaving each intersection per hour app ears beside each road. Fifth and Sixth Avenues can handleup to 2000 vehicles per hour without causing congestion, w hereas the maximum capacity of each ofthe two streets is 1000 vehicles per hour. The flow of traffic is controlled by traffic lights installed ateach of the four intersections.1. Set up the system of equations thatwould model this pr ob lem.2. Solve the system of equations and writethe answer in parametric form . Placerestrictions on the parameter.3. Find two possible flow patters thatwould ensure that there is no traffic con-gestion.500350120011004005501400xyzw15004th st.5th st.5th Av.6th Av.Math 141-copyright Joe Kahlig, 10B Page 5Example: Give the solution for this problem.x = the number of small drinksy = the number of medium drinksz = the number of large drinksx y z1 0 -0.5 -30 1 4590 0
View Full Document
Unlocking...