Section 2.1 Systems of Linear Equations: An Introduction

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1 Section.1 Systems of Linear Equations: An Introduction Three Possible Outcomes for two lines, L 1 and L a) L 1 and L can cross at one and only one point (Unique solution) b) L 1 and L are parallel and coincide (Infinitely many solutions) c) L 1 and L are parallel and never cross (No solution) Unique Solution: x +3y = x + y = Infinitely Many Solutions: x +3y =3 4x +y =

2 No Solution: x +3y = x 3y = Example 1: Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution. (If there are infinitely many solutions, express x and y in terms of the parameter t.) 4 x 3 y = 1 4 x + 3 y =1 Example : Determine the value of k for which the system of linear equations below has no solution. 3x y =3 9x + ky = Fall 017, Maya Johnson

3 Applications of Systems of Equations For the following two examples, we will setup but not solve the resulting system of equations. Example 3: The Johnson Farm has 00 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $44 and $8 per acre, respectively. Jacob Johnson has $1, 00 available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant? (Let x and y denote the number of acres of corn and wheat, respectively.) 3 Fall 017, Maya Johnson

4 Example 4: The management of a private investment club has a fund of $100, 000 earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk (x), medium risk (y), and low risk (z). Management estimates that high risk stocks will have a rate of return of 1%/year; medium risk stocks, 10%/year; and low risk stocks, %/year. The investment in low risk stocks is to be twice the sum of the investments in stocks of the other two categories. If the investment goal is to have a rate of return of 9% on the total investment, determine how much the club should invest in each type of stock. (Assume that all the money available for investment is invested.) 4 Fall 017, Maya Johnson

5 Section. Systems of Linear Equations: Unique Solutions Augmented Matrices The system of equations x +4y 8z = can be represented as the following matrix 3x 8y +z =7 x 7z = Example 1: What value is in row 1, column of the above matrix? Example : Find the augmented matrix for the following system of equations. 9x +y 10z =11 4x 1y +17z =37 x y =4 Example 3: Find the system of equations for the following augmented matrix Fall 017, Maya Johnson

6 RowReduced Form of a Matrix 1. Each row consisting entirely of zeros lies below all rows having nonzero entries. The first nonzero entry in each (nonzero) row is a 1 (called a leading 1). 3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading 1intheupperrow. 4. If a column in the coe cient matrix has a leading 1, then the other entries in the column are zeros. Example 4: Which of the matrices below are in rowreduced form? Row Operations 1. Interchange any two rows.. Replace any row by a nonzero constant multiple of itself. 3. Replace any row by the sum of that row and a constant multiple of any other row. Unit Column Acolumninacoe cientmatrixiscalledaunit column if one of the entries is a 1 and the other entries are zeros. Note: If you transform a column in a coe cient matrix into a unit column then this is called pivotting on that column. Notation for Row Operations Letting R i denote the ith row of a matrix, we write: Operation 1. R i $ R j Interchange row i with row j. Operation. cr i to mean: Replace row i with c times row i. Operation 3. R i + ar j to mean: Replace row i with the sum of row i and a times row j. Fall 017, Maya Johnson

7 Example : Pivot the matrix below about the entry in row 1, column 1 " # The GaussJordan Elimination Method 1. Write the augmented matrix corresponding to the Linear system.. Interchange rows (Operation 1), if needed, to obtain an augmented matrix in which the first entry in the first row is nonzero. Then pivot the matrix about this entry. 3. Interchange the second row with any row below it, if needed, to obtain an augmented matrix in which the second entry in the second row is nonzero. Then pivot the matrix about this entry. 4. Continue until the final matrix is in rowreduced form. Example : Solve the following system of linear equations using the GaussJordan elimination method. a) x +y =1 x +8y =10 7 Fall 017, Maya Johnson

8 b) x +y =4 3x +y = c) x 1 + x x 3 =3 3x 1 +x + x 3 =8 x 1 +x +x 3 =4 Example 7: A person has four times as many pennies as dimes. If the total face value of these coins is $1., how many of each type of coin does this person have? k :ky y:a: ahead 8 Fall 017, Maya Johnson

9 bedroom bedroom Example 8: Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of onebedroom units and two and threebedroom townhouses. A total of 18 units is planned, and the number of family units (two and threebedroom townhouses) will equal the number of onebedroom units. If the number of onebedroom units will be 3 times the number of threebedroom units, find how many units of each type will be in the complex. = X # of one bedroom y # of two = z = # of three bedroom units units bedroom units " total 18 units " xtytz = 18 " # of family units equals # of one bedroom units " yt Z a " # of one units is 3 times three units " = zz System of Linear Equations : X ty tz = 18 X ty t Z = 18 y + Z = = ZZ + y 3Z t z = O = 0 ftty±h : :d F.. or s sw?jbyyommuunn 8 three bedroom units 9 Fall 017, Maya Johnson