Notes #3- Date: 7.1 Solving Systems of Two Equations (568) The solution to a system of linear equations is the ordered pair (x, y) where the lines intersect! A solution can be substituted into both equations to make true statements. Consistent solution(s) Inconsistent Ø Dependent ( ) Independent (1 or Ø) These lines coincide so they are a.k.a coincident. Substitution 1. Solve an equation for one of the variables (if necessary).. Substitute the expression for the variable and solve. 3. Substitute into either equation to get the other value. y = x + 3 1. Not necessary. 3x + y = -1. 3x + (x + 3) = -1 3x + 4x + 6 = -1 7x + 6 = -1,7x = -7, x = -1 3. y = (-1) + 3 = - + 3 = 1 Ans: (-1, 1) Ex.1 Solve using substitution: a) 3x + y = 14 b) -3x + y = 7 x y = 10-6x + y = -8
You do not have to substitute for a single variable (you can use an expression). Ex. Solving nonlinear systems: a) y = x 1 b) x + y = 6 x+ y x+ y + = x x + y = 13 ( ) 6 Graphing Ex.3 Graph and determine the point of intersection.: a) 3x + y = 4 b) 3x + y = 6 x y = 6 y = ln(x 1) Linear Combination or Elimination 1. Arrange like terms in columns (if necessary).. Multiply so that one set of variables are opposites. 3. Add and then solve. 4. Substitute into either equation to get the other value. y = x + 3 1. -x both sides -x + y = 3 3x + y = -1 3x + y = -1. Eliminate y if we multiply eq.#1 by - -(-x + y = 3) = 4x y = -6 3x + y = -1 3. 7x = -7 x = -1 4. y = (-1) + 3 = - + 3 = 1 Ans: (-1, 1)
Ex.4 Solve using elimination: a) Find the product of the x & y coordinates of the solution: 3x y = 4 x + 5y = - b) x 4y = 8 x y = 4 Ex.5 Three gallons of a mixture is 60% water by volume. Determine the number of gallons of water that must be added to bring the mixture to 75% water. Ex.6 A car radiator contains 10 quarts of a 30% antifreeze solution. How many quarts will have to be replaced with pure antifreeze if the resulting solution is to be 50% antifreeze?
Ex.7 A machine takes 3 minutes to form a bowl and minutes to form a plate. The material costs $0.5 for a bowl and $0.0 for a plate. How many bowls and plates were made if the machine ran for 8 hours and $44 was spent on materials? Ex.8 A company determines its supply and demand equations for a product to be: S: p = x + 4x + 40 D: p = 50 3x Find the equilibrium point in terms of x (thousands of the item) and p (the price). What does it mean? What relationship does the company want to have between the supply and the demand? Summary:
Notes #3- Date: 7. Matrix Algebra Day 1 (579-583) A = 1 1 0 5 3 Order (dimensions) 3 (rows) r c (columns) If r = c then it is a square matrix entry/element a 3 = - (row 3, column ) Ex.1 Determine the order: Is it a square matrix? Identify: B = b 33 = b 4 = 5 1 4 4 8 0 6 3 1 1 In addition & subtraction, matrices must have the same order. Add/subtract the corresponding elements. You multiply all elements of a matrix by a scalar multiple (think distributive). Ex. 0 4 8 7 1 0 = Scalar Multiple Identities: Addition: 3 + 0 = 3 Multiplication: 3 1 = 3 Matrix Addition: 14 14 + = 1 0 1 0
Matrix Multiplication (not commutative AB BA): Ex.3 Find the product AB: B A 5 1 4 4 1 4 8 0 6 5 8 0 i 3 1 1 3 3 4 Must be = to Order of answer 4. ()+4(4)+(-1)(-3) 10 + 3 + (-1) = + 0 + = Move me! Move me! Move me! 4+16+3=3 41 11 1 4 13 14 1 3 4 Ex.4 Find the product: 1 0 1 3 1 1 1 0 i 3 Ex.5 Redo Ex.3 & Ex.4 on the graphing calculator. List the steps in your notes.
Identity: Inverse: 3 + 0 = 3 3 + (-3) = 0 (opposite) 3 1 = 3 3 1 3 = 1 (reciprocal) 1 0 0 1 3 4 + = 0 0 3 4 3 1 3 1 1 i = 1 1 1 0 0 3 4 + = 3 4 0 0 3 1 1 0 1 i = 0 1 Inverse not a negative exponent 3 1 1 1 Ex.6 Verify: and A-1 A = 1 = 3 Determinant: If A = a c Ex.7 Find the value of: a) 7 9 3 6 b d, then det(a) = A = ad bc 4 b) det 3 7
Ex.8 Find the inverse of a : A = 3 1 1 1. Find det(a), no A -1 if det(a) = 0.. Exchange a & d. 3. Opposite signs on b & c. 4. Multiply by the reciprocal of det(a). Ex.9 Find the inverse of: 4 1 6 Summary:
Notes #3- Date: 7. Matrix Algebra Day (584-589) A matrix that has a nonzero determinant is called a nonsingular matrix (which means it has an inverse). The minor (short for minor determinant) of an element of an nth order determinant can be found by deleting the row and column containing the element. A = 4 1 0 3 6 5 4 The minor of a 11 is 3. 5 4 The cofactor of an element is (-1) i+j M ij. So the cofactor of a 11 is (-1) 1+1 (-1 10) = (-1) (-) = 1(-) = -. Another way to deal with the signs is to follow the pattern that a 11 is + and adjacent elements (horizontal & vertical) are the opposite sign. + + + + + To evaluate a determinant of the nth order, multiple the elements of a row or column by their corresponding cofactors: -(+) 3 5 4 + 0(-) 4 1 4 + (-6)(+) 4 3 = 1 5
Ex.1 Find the value of: 4 3 a) 5 1 0 1 11 b) det 0 1 0 5 6 0 3 4 5 6 0 3 0 4 Lattice method: 4 3 4 5 1 0 5 1 1 11 1 -(6 + 0 + 110) + (44 + 0 + -15) -116 + 9 = -87 Ex. Find the inverse of a : A = 3 1 1 Adjoin the identity and perform row ops to get the identity on the left: 3 1 1 0 1 0 1 If you get the 0s first you will have less fractions to deal with. Switching rows is also a row operation. Row operations are basically linear combinations.
Same as by. Ex.3 Find the inverse: 0 1 1 1 1 3 0 Cramer s Rule: ax + by = c dx + ey = f D x = x = D c f a d b e b e a c Dy d f y = = D a b d e If you get 0 0 you have coincident lines (planes) & # 0 is Ø. Ex.4 Use Cramer s Rule to solve: 5x 3y = 4 x + 4y = 1 Ex.5 Use Cramer s Rule to solve for z: a) x y z = 4 b) x y + z = 8 4x + y + z = 7 x + 3y z = - x + 3y 4z = -1 3x y 9z = 9 Summary:
S R1 & R - R1 + R3 R3 R -5R + R3 -.5 R3 -R3 + R1 R + R1 0 1 1 0 0 1 1 1 0 1 0 3 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 3 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 5 0 1 1 1 1 0 1 0 0 1 0 1 0 5 0 1 1 1 1 0 1 0 0 1 0 1 0 0 1 5 4 1 1 0 5 5 0 1 0 1 0 0 1 5 4 0 0 3 0 0 0 1 0 5 0 1 0 5 0 1 0 4 0 0 0 1 0 1 0 5 0 10 10 5 0 5 0 1 0 0 10 8 4 0 0 1 5 4 1 1 1 0 1 0 1 1 0 5 5 1 0 0 3 3 1 0 1 0 1 0 0 1 5 4 A -1 = 3 3 1 1 5 4
Notes #3- Date: 7.3 Multivariate Linear Systems & Row Operations (594) Triangular Form (leading coefficients = 1): Ex.1 Use back substitution: 1x y z = 7 1y z = -7 1z = 3 It is easier to get the 0s first, if getting a 1 leads to fractions. Verify with sub or elimination. Gaussian Elimination: Interchange rows or use elimination to change rows. Ex. x + y z = 0 x y + z = -3 -x + y 3z = 7 1. Exactly One Solution. Infinite Solutions 3. No Solutions x y + z = 8 x + 3y z = - 3x y 9z = 9 Row Echelon Form (REF): Ex.3 3x 6y + 3z = 7 x + y + 4z = x y + z = 4 Augmented Matrix 1 1 1 8 3 1 3 9 9 3 4
Gauss-Jordan Elimination (Reduced REF): Ex.4 3x + y + z = 5 Ex.5 x + y z = 1 x y + z = 4 3x + 3y + z = 4 x + 3y z = -3 x y z = 0 Ex.6 x + y 3z = -1 Ex.7 Nonsquare Systems: y z = 0 x y z = -5 -x + y = 1 x + y + z = 5 A is called the coefficient matrix. Inverse matrices can be used to solve systems: x = 4 is solved by using inverses: 1 x = 1 4, x = 4x + y = 4 4 1 x 4 Ex.7 6x y= 3 6 i = y 3
Ex.8 Solve Ausing -1 A inverse X = Amatrices: -1 B a) x + y + z = -4 b) x + y + z + w = 4 x y + z = -4 x y + z = 0 x + 3y z = -7 3x + y + z w = 6 x y z + w = -1 (-3, -1, 1) Ex.9 You borrow $10,000; some at 18%, 15% and 9%. You borrow 3 times as much at 15% as you borrow at 18% and the interest is $144.5. How much each? Ex.10 Determine the quadratic function that contains the points: (1, 3), (, 10) & (-, -6). y = x + 4x Summary:
Notes #3- Date: Not unique: + 1 3 6 7.4 Partial Fraction Decomposition (608) I. Intro: Ex.1 1 1 1 1 3 3 + = + = + = 5 3 3 3 6 6 6 How can we get back? Unique? Ex. 3 + x 4 x 3 5x+ 1 + = = ( x 4)( x+ 3) PFD: writing rational expressions as the sum of simpler ones. II. Distinct (no repeated) linear factors in the denominator. Step #1: Factor the denominator. Step #: Write the factors as a sum. Ex.3 5x 10 x + 3x = = A B x 1 + x+ Step #3: How do we solve rational equations? Multiply by the LCD. Step #4: Solve the resulting system of equations.
16 Ex.4 t 4t III. Repeated Linear Factors x+1 A A = + + + 1 n n ( x 7) x 7 ( x 7) ( x 7) A n Ex.5 x+ 3 x x + x Strategic values of x may help you solve the system. See Ex.3. IV. Irreducible Distinct Quadratic Factors x+ 1 A Bx+ C = + x+ x + x+ 1 ( x+ )( x + x+ 1) Ex.6 8x + 5x+ 18 3 x 8
V. Irreducible Repeated Quadratic Factors x+ 1 A Bx+ C Dx+ E = + + ( x+ 1 )( x + x+ 1 x+ x + x+ ) ( x + x+ 1) Ex.7 3 x x x + 5 ( x + 3) Summary:
Notes #3- Date: 7.4 Partial Fractions Day (613) I. Intro: Ex.1 Write the terms for the PFD (don t solve): x 3x ( + + 5)( + ) xx x x 3 = Ex. Decompose: 3 3x 6x + 7x ( x x+ ) Ex.3 Use division to rewrite in the form 4 3 x 3x + x 5 x x+ 1 rx ( ) qx ( ) + : dx ( )
II. Improper Fractions (Degree of N > D): Ex.4 Use division to rewrite in the form rx ( ) qx ( ) + and dx ( ) rx ( ) then find the PFD of dx ( ) : f(x) = 4 3 9 x + x x+ 6 xx ( + ) Where is the low battery warning? Compare the rational f(x) graph to the terms of the PFD. rx ( ) Ex.5 Use division to rewrite in the form qx ( ) + and dx ( ) rx ( ) then find the PFD of dx ( ) : 3 3x + 4x + x+ 6 x 1 Summary:
Notes #3- Date: 7.5 Systems of Inequalities in Two Variables (617) The graph of a linear inequality is a half-plane. More lead? Number Line Coordinate Plane < & > Open dot Dotted line < & Closed (solid) dot Solid Line Use a test point for each inequality. Ex.1 Graph (are they bounded or unbounded): a) x < 4, y -3 b) x 1, x < 4, y -1 and x y 4 & y 1 x + 1. Ex. Write a system of inequalities:
Ex.3 Solve the system of inequalities. 1 a) y> x b) y> x 1+ 3 y < -x x + x + y < 8 Linear Programming: Constraints: a system of inequalities Feasible Region: solution set Objective Function: an equation applied to a feasible region Optimal Solution: maximum or minimum of the solution set Fundamental Theorem of Linear Programming: the maximum or minimum always occurs at a vertex. Ex.4 Mr. Jones is baking as many cakes & pies as possible using the 18 eggs & 15 cups of milk he has on hand. His cake recipe requires eggs and 1 cup of milk and his pie recipe requires 1 egg and 1.5 cups of milk. a) Find the constraints and graph the feasible region. b) Write the objective function and determine how many cakes & pies should be made.
Ex.5 It takes hrs to manufacture a pair of skis and 1 hr for a snowboard. The finishing time for both is 1 hr. The maximum time available for manufacturing is 40 hrs and for finishing is 3 hours each week. The profit for a pair of skis is $70 & the profit for a snowboard is $50. The manufacturer must produce at least 8 snowboards every week because of a contract with a sporting goods store. a) Find the constraints and graph the feasible region. b) Write the objective function & find the maximum profit. Summary: