Math 1314 Week #14 Notes

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1 Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter, our focus will be solving systems of linear equations. If we are solving a system of two linear equations in two variables, there are three possible types of systems we could have: Case # Case # Case #3 Consistent System Inconsistent System Consistent System The equations are The equations are The equations are independent. Independent. dependent The solution is the The lines do not Every point on the where the lines intersect, so there line is a solution. We intersect. In this case, is no solution, { }. write the solution as: the solution is (, ). The lines are different, {(x, y) ax + by = c} The lines have but the lines do have The lines are the different slopes. the same slope. same or coincident. The graph of the equation in the form ax + by + cz = d is a plane in three dimensional space. The points in three-dimensional space have three coordinates, x, y, and z. We will list them in alphabetical order: (x, y, z). In solving a system of three linear equations in three variables, we are not looking at intersecting lines, but intersecting planes. Here are some possible types of systems we could have:

2 Type : The solution is a point: Type : The solution is a line: Two planes are the same.

3 Type 3: There is no solution: Two planes are the same. Type : Solution is a plane: All three planes are the same. When dividing a polynomial by a factor of x a, we have the option of using synthetic division rather than using long division. This allows us to work with just the numbers rather than having to keep track of the variables as well. We can do the same thing when we are solving systems of equations. We use matrices to represent the system of equations.

4 System of Equations Matrix 5x 3y = 3 [ ] Two rows x + y = 7 7 Three columns Row # Row # Row #m a + b c = 6 6 Three rows 7a + b 8c = 7 8 Four columns a 7b + 5c = 7 5 Since the first matrix has two rows and three columns, it is called 3 matrix, while the second matrix is 3 matrix. We always classify our matrices by the number of rows and then the number of columns. Definition An m n matrix is a rectangular array of numbers with m rows and n columns Column Column Column # # #n a a a n a a a n a m a m a mn [ ] Each entry a ij in the matrix has two indices indicating the row i (the first number) and the column j (the second number) of where the entry is located. Hence, a 3 represents the number in the th row, 3 rd column of the matrix. In using synthetic division, it was important to have the dividend in order of descending powers and if there was a term that was missing, we needed to use a as the coefficient for that term. In writing an augmented matrix to represent a system of linear equations, we need have the variables in order on the left side of the equation and the constants on the right side. Also, if there is a variable missing, we will need to use as the coefficient. Row Operations There are three basic row operations we can perform on a specific row of a matrix that will yield new matrix that represents a system of linear equations that is equivalent to the system of linear equations represented

5 by the original matrix. We will use capital letters in the notation to represent the new entries in the row and lower case letters to represent the original entries in a row. ) Interchange any two rows (example: R 3 R means interchange row 3 and row ). ) Replace a particular row by a nonzero multiple of that row (example: R = 3r means multiply row by 3 and make the answer the new row ). 3) Replace a particular row by the sum of that row and a nonzero multiple of any other row (example: R = r + 5r 3 means multiply row 3 by 5 and add to row. Make the answer the new row ). Definition A matrix is in Row Echelon Form when all of the following are true: ) Reading from left to right, the first nonzero entry in each row is. This is referred to as the leading entry. ) Reading from left to right, the first nonzero entry of a row (leading entry) occurs to the right of the first nonzero entries of all the proceeding rows. 3) Any rows that contain all zeros will be below all of the rows that have nonzero entries. Example: 7 This matrix is in row echelon form. Now, write the system of equations that correspond to this matrix. ) x + y z = ) y + z = 7 3) z = Finally, use back-substitution to solve: Replace z by in equation # and solve for y: y + ( ) = 7 y 8 = 7 y = Now, replace z by and y by in equation # and solve for x: x + () ( ) = x + + = 3 x + = x = 3 Thus, the solution is ( 3,, ). The process of getting the matrix in row echelon form and using back-substitution to solve a system of linear equations is called Gaussian Elimination.

6 Solve a System of Linear Equations Using Gaussian Elimination ) Write the matrix that represents the system of linear equations. ) Use elementary row operations to rewrite the matrix in row echelon form. This is done by: a) Use row operations to get an entry of in the top left corner and then get zeros in the rest of the column below that. b) Next, use row operations to get a leading entry of in the next row and get zeros in the rest of the column below that. Be sure that the leading entry is to the right of all leading entries in the proceeding rows. c) Repeat step #b until you get through the bottom row or until the rest of the rows have all zero entries. 3) Write the new system of linear equations that corresponds to the matrix found in part and use back-substitution to solve. A matrix is in Reduced Row Echelon Form if it is in Row Echelon Form and if all the entries aside from the leading entry of in each row are. Example: 3 In this form, we can immediately find the solution to this system since the correspond system of equations is: x = 3; y = ; z = The process of getting a matrix into reduced row echelon form is call Gauss-Jordan elimination. Solve a System of Linear Equations Using Gauss-Jordan Elimination ) Write the matrix that represents the system of linear equations. ) Use elementary row operations to rewrite the matrix in row echelon form. This is done by: a) Use row operations to get an entry of in the top left corner and then get zeros in the rest of the column below that. b) Next, use row operations to get a leading entry of in the next row and get zeros in the rest of the column above and below that. Be sure that the leading entry is to the right of all leading entries in the proceeding rows. c) Repeat step #b until you get through the bottom row or until the rest of the rows have all zero entries. 3) Write the new system of linear equations that corresponds to the matrix and write down the answer.