Food I Food II Food III Vitamin A Vitamin C Calcium

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1 Math 141 Linear systems and Matrices Determine whether each system of linear equations has (a one and only one solution, (b infinitely many solutions, or (c no solution. Find all solutions whenever they exist x 6y = 8 10x 12y = x + 15y = -3 4x 6y = (5/4x (2/3y = 3 (1/4x + (5/3y = Determine the value of k such that the following system of linear equations has a solution, and then find the solution: 2x + 3y = 2 x + 4y = 6 5x + ky = Nutrition A dietician wishes to plan a meal around three foods. The meal is to include 8800 units of vitamin A, 3380 units of vitamin C, and 1020 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table: Food I Food II Food III Vitamin A Vitamin C Calcium Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements. 6.- Manufacturing Production Schedule Ace Novelty manufactures Giant Pandas, Saint Bernards, and Big Birds. Each Giant Panda requires 1.5 yd 2 of plush, 30 ft 3 of stuffing, and 5 pieces of trim; each Saint Bernard requires 2 yd 2 of plush, 35 ft 3 of stuffing, and 8 pieces of trim; and each Big Bird requires 2.5 yd 2 of plush, 25 ft 3 of stuffing, and 15 pieces of trim. If 4700 yd 2 of plush, 65,000 ft 3 of stuffing, and 23,400 pieces of trim are available, how many of each of the each of the stuffed animals should the company manufacture if all the material is to be used? Give two specific options.

2 7.- A manufacturer of women s blouses makes three types of blouses: sleeveless, short-sleeve, and long-sleeve. The time (in minutes required by each department to produce a dozen blouses of each type is shown in the following table: Sleeveless Short-Sleeve Long-Sleeve Cutting Sewing Packaging The cutting, sewing, and packaging departments have available a maximum of 80, 160, and 48 labor-hours, respectively, per day. How many dozens of each type of blouse can be produced each day if the plant is operated at full capacity? In Exercises below, solve the system of linear equations, using the Gauss-Jordan elimination method x + 6y = 8 3x - 2y = -7 x + 3y = x y + 2z = 5 x y + 2z = 1 5x 2y + 4z = x 1 x 2 + x 3 = -4 3x 1 (3/2 x 2 + (3/2 x 3 = -6-6x 1 + 3x 2 3x 3 = x 1 + 6x 2 5x 3 = 5 x 1 + 3x 2 + x 3 + 7x 4 = -1 3x 1 + 9x 2 x x 4 = x 1 2x 2 + x 3 = -3 2x 1 + x 2 2x 3 = 2 x 1 + 3x 2 3x 3 = 5

3 In Exercises below, given that the augmented matrix in row reduced form is equivalent to the augmented matrix of a system of linear equations, (a determine whether the system has a solution and (b find the solution or solutions to the system, if they exist

4 17.- The Cinema Center consists of four theaters: Cinemas I, II, III, and IV. The admission price for one feature at the Center is $4 for children, $6 for students, and $8 for adults. The attendance for the Sunday matinee is given by the matrix A = Cinema I Cinema II Cinema III Cinema IV Ch St Ad ( a.- Write a column vector B representing the admission prices b.- Compute AB, the column vector showing the gross receipts for each theater c.- Find the total revenue collected at the cinema Center for admission that Sunday afternoon Consider the system -2x + 3y + z = -5 x y z = 1-2x 2y + 2z = -3 a.- Write a matrix equation (in the form AX = B that is equivalent to the system of linear equations. b.- Solve the system, and write the solution typing explicitly the theory of inverse matrix Find the values of the variables ( x y 2 ( z 4 2 = (

5 20.- Three network consultants, Alan, Maria, and Steven, each received a year-end bonus of $10,000, which they decided to invest in a 401(k retirement plan sponsored by their employer. Under this plan, employees are allowed to place their investments in three funds: an equity index fund (I, a growth fund (II, and a global equity fund (III. The allocations of the investments (in dollars of the three employees at the beginning of the year are summarized in the matrix. A = I II III Alan ( Maria Steven The returns of the three funds after 1 yr are given by B = I ( 0.15 II 0.21 III 0.14 Determine which employee realized the best return on his or her investment for the year in question? 21.-Matrix A shows the pounds of nails (n, screws (s, and bolts (b needed to build a cottage (C, townhome (T, and villa (V. Matrix B shows the number of cottages, townhomes, and villas that will be built in the subdivisions of Aggie (A, Maroon (M, and White (W. Which product matrix has meaning; interpret its meaning? n s b A M W Cottage ( Cottage ( A = Townhome B = Townhome Villa Villa a. A T B yields the number of pounds of nails, screws and bolts needed to build in each subdivision b. B -1 A yields the number of pounds of nails, screws and bolts needed to build each cottage, townhome and villa c. A B T yields the number of pounds of nails, screws and bolts needed to build each subdivision d. B A T yields the number of pounds of nails, screws and bolts needed to build each cottage, townhome and villa e. A -1 B yields the number of cottages, townhomes and villas in each subdivision.

6 22.- Determine if the next couple of matrices are inverse to each other a.- A = ( B = ( b.- A = ( B = ( c.- A = ( B = (

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8 Concepts to Know The Cartesian Coordinate System x and y axis (label your scales on graphs Ordered pairs (points: (x, y Origin: (0, Straight Lines Slope = m = (y2 y1 / ( x2 x1 Positive slope: line rises from left to right Negative slope: line falls from left to right Zero slope: Horizontal line Undefined Slope: Vertical line Equations of Lines Pt-Slope Form: y y1 = m(x x1 Slope-Int Form: y = mx + b General Form: Ax + By + C = 0 Horizontal line: y = a Vertical line: x = b Intercepts x-intercept = point where line crosses x-axis:(#,0 y-intercept = point where line crosses y-axis:(0,# Parallel Lines = m1 = m2 (same slopes/diff. Y-int Perpendicular Lines = m1m2 = 1 (neg. reciprocal slopes Linear Functions and Math. Models Functions Domain Range Independent Variable Dependent Variable Linear Functions

9 Linear Depreciation: V (t = mt + b m = rate of depreciation b = value of asset at time = 0 Scrap Value= lowest value asset attains Linear Cost, Revenue and Profit Cost: C(x = cx + F where c is the cost to produce each unit and F is the fixed costs Revenue: R(x = sx where s is the selling price of each unit Profit: P(x = R(x C(x Linear Supply and Demand S(x = p = mx + b D(x = p = mx + b ** All points on supply and demand curves are of the form (x, p =(quantity, price!! ** Intersection of Straight Lines Break-Even Point R(x = C(x (or P(x = 0 to find quantity x = break-even quantity y = break-even revenue/cost Equilibrium Point Supply = Demand x = equilibrium quantity y = equilibrium price The Method of Least Squares/Linear Regression Be able to use LinReg on your calculator to find the least-sq. line Correlation coefficient (r - determines the amount of the data explained by the line (Want r close to 1 Be able to predict values, using the least-sq. Line Systems of Linear Equations Two linear equations = three cases Unique Solution (intersecting lines = m1 6= m2 No Solution (parallel lines = m1 = m2 and b1 6= b2 Infinitely Many Solutions (same line = m1 = m2 and b1 = b2

10 General solution (parametric solution Specific solutions Setting up systems of equations Matrices Size (dimension: mxn (m = # rows, n = #columns Matrix elements: aij (element in row i and col j Equality = all corresponding entries equal Addition/Subtraction (matrices must be the samesize Matrix Transpose (A T : switch rows and cols Scalar Multiplication: multiply every entry by thescalar 2.2/2.3 - Solving System of Equations Gauss-Jordan (GJ Elimination (Row Operations Interchange any two equations Multiply an eqn by a non-zero constant Add a multiple of one eqn to another Row-Reduced Echelon Form All zero rows must be below all non-zero rows The first non-zero entry in each row is 1 (lead-ing 1 In any two successive (non-zero rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row If a column contains a leading 1, then the other entries in that column are zeros RREF on your calculator Solving a system Put system in nice form Place nice system in an augmented matrix Use RREF on your calculator to reduce your system Read system (eqns from reduced system Find soln Multiplication of Matrices The # of cols in the left matrix must equal the # of rows in the right matrix. (Inner dimensions equal. If A is (mxn and B is (nxr, then AB is (mxr.

11 (Outer dimensions give answer size. ORDER IS IMPORTANT! Know how to multiply matrices by hand. Identity Matrix: Square matrix (# rows = # cols with 1 s along the diagonal (from upper lft to lower right and 0 s elsewhere Be able to represent a system of equations as a matrix equation: AX = B Put system in standrad form A = coefficient matrix X = variable matrix B = constant matrix Inverse of a Square Matrix Matrix must be square Not all matrices have an inverse ( singular means no inverse Inverse of A = A 1 AA 1 =A 1 A = I A system in form AX = B = X = A 1 B if the inverse exists (know how to solve a matrix equation