Solving Systems of Linear Equations Case 1: One Solution Case : No Solution Case 3: Infinite Solutions Independent System Inconsistent System Dependent System x = 4 y = Classification by Number of Solutions At Least 1 Solution Consistent System No Solution Inconsistent System Reach a Contradiction ex) 0 = Exactly 1 Solution Independent System x = ex) y = 4 Infinite Number of Solutions Dependent System Reach an Identity ex) 0 = 0
Solving Systems of Equations Case 1: One Solution: Independent System Graphing Method Addition / Elimination Method x 3y = 4 8x 1y = 8 3x + 4y = 0 3 9x + 1y = 60 17x = 68 x = 4 3x + 4y = 0 3(4) + 4y = 0 4y = 8 y = Conclusion : x = 4, y = Case : No Solution: Inconsistent System Substitution Method y = x + y = (x + ) x y = 4 x = 4 ( ) x x + = 4 x x = 4 A Contradiction = 4 y Conclusion: No Solution Case 3: Infinite Number of Solutions: Dependent System Addition / Elimination Method x + y = 6 x + y = 6 1 x + y = 6 x = 6 y x + y = 6 ( 1) x y = 6 0 + 0 = 0 An Identity 0 = 0 Conclusion : Dependent System All points on the line: x + y = 6
Solve: Using your TI-83 Graphing Calculator to Solve Systems of Linear Equations System of Equations x + y = 4 3x y + z = 4x + y 3z = 19 Format the Matrix nd MATRX EDIT 1: [A] [Enter] MATRIX [A] 3 4 Corresponding Augmented Matrix 1 0 4 3 1 1 4 3 19 3 rows 4 columns Enter the Dimensions of the Matrix (Rows Columns) Enter the Entries of the Matrix Enter the Coefficients of x, y, z and the Constant. 1 [ENTER] [ENTER] 0 [ENTER] 4 [ENTER] 3 [ENTER] 1 [ENTER] 1 [ENTER] [ENTER] 4 [ENTER] [ENTER] 3 [ENTER] 19 [ENTER] nd QUIT to Exit the Matrix Mode [CLEAR] to Clear the Screen Transform the Matrix to Reduced Row Echelon Form nd MATRX Math B: rref( Scroll Down to reduced row echelon form [ENTER] nd MATRX NAMES 1: [A] 3 4 [ENTER] [ENTER] Interpret the Matrix and State the Solution x y z 1 0 0 0 1 0 1 0 0 1 3 x = y = 1 z = 3
Matrix Row Transformations For any augmented matrix of a system of linear equations, the following row transformations will result in a matrix of an equivalent system (i.e. has the same solution set). 1. Interchange / Swap any two rows.. Multiply any row by a nonzero real number. 3. Add a multiple of one row to another row.
Gaussian Elimination 3x + 5y z = 7 x + y + z = 1 x + y + 11z = 7 3x + 5y 1z = 7 1x + 1y + 1z = 1 x + 1y + 11z = 7 3 5 1 7 1 1 1 1 1 11 7 1 1 1 1 3 5 1 7 1 11 7 3 1 1 1 1 0 4 4 1 11 7 1 1 1 1 1 0 1 1 11 7 1 1 1 1 0 1 0 1 9 9 1 Swap R 1 and R R 3R 1 R = + R 1 R = R + R = R 3 1 3 x y z 1 1 1 1 0 1 0 0 7 7 1 7 1 1 1 1 0 1 0 0 1 1 1x + 1y + 1z = 1 0x + 1y z = 0x + 0y + 1z = 1 x + y + z = 1 y z = z = 1 R3 = R + R 1 3 R3 = R 7 3 1 1 1 1 0 1 0 0 1 1 Row Echelon Form 1z = 1 z = 1 Using Back-Substitution: x =, y = 0, z = 1 1y z = y (1) = y = 0 1x + 1y + 1z = 1 x + 1(0) + 1(1) = 1 x = Without Using Back-Substitution: 1 1 1 1 0 1 0 0 1 1 Row Echelon Form 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 x y z 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 1 R = R3 + R R1 = 1R3 + R1 R1 = 1R + R1 x = y = 0 z = 1 Reduced Row Echelon Form
Cramer s Rule a x b y c 1 1 1 a x b y c a1 b1 c1 a b c c b a c 1 1 1 1 D c b D a c, for 0 D a b D a b x y x y D 1 1 1 1 a b a b a x b y c z d 1 1 1 1 a x b y c z d a x b y c z d 3 3 3 3 a b c d 1 1 1 1 a b c d a b c d 3 3 3 3 d b c a d c a b d 1 1 1 1 1 1 1 1 1 d b c a d c a b d D d b c D a d c D a b d,, for 0 D a1 b1 c1 D a1 b1 c1 D a1 b1 c1 a b c a b c a b c x 3 3 3 y 3 3 3 z 3 3 3 x y z D a b c a b c a b c 3 3 3 3 3 3 3 3 3
Partial Fraction Decomposition I f( x) must be a proper fraction. gx ( ) That is, the degree of the numerator must be less than the degree of the denominator. If not, use long division to divide the denominator into the numerator. II Linear Factors A) Distinct Linear Factors ( ) ( ) ( ) f x = A + B ax + b cx + d ax + b cx + d B) Repeated Linear Factors f ( x) A B = + ( ax + b) ax + b ( ax + b) Ex) f( x) A B C D E F = + + + + + 3 3 x x x x x 5 x x + 7 ( 5) ( + 7) x x ( 5) III Quadratic Factors A) Distinct Quadratic Factors f( x) Ax + B Cx + D = + ( ax + bx + c )( ax + bx + c ) ( ax + bx + c ) ( ax + bx + c ) 1 1 1 1 1 1 B) Repeated Quadratic Factors f ( x) Ax + B Cx + D = + ( ax + bx + c ax + bx + c ) ( ax + bx + c) Ex) f( x) A B C Dx + E Fx + G = + + + + x x x x ( 5) ( + 7 x x x 5 x + 7 ) ( + 7)
Partial Fraction Decomposition SETUP the Partial Fraction Decomposition of the following rational expressions. Do NOT solve the resulting equations, that is do NOT find the values of A, B, etc. Note: FACTOR the denominator (if possible) before decomposing the fraction. 1) x 3x 4 4x 1 ) 5x + 3x 4 ( x + 1)( x 5)( x + x + 7) 3) x ( ) 8x 1 x 1 ( x + x + 5) 4) x 8x 1 3 x ( x + ) ( x + x + 5) 5) 3 x x + 3x + 9 3 ( x + 5x + 4 ) ( x + 7 ) ( x + 8 ) 6) 3 x x + 3x + 9 ( + 7 ) ( + 8 ) 3 x x x
MAC1140 Sample Test 4 Name 1) Solve the following system of linear equations by each of the indicated methods. Show your work. x + 3y = 1 y = x 4 A) Graphing Method B) Substitution C) Addition Elimination ) Use your calculator to solve the following system of linear equations. x 3y + z = x + y z = 10 5x 3y + 4z = 5
p MAC1140 Pre-Calculus Algebra Sample Test 4 Falzone page of 3 3) Solve the following system of linear equations algebraically. Use Gaussian Elimination to reduce the following system of linear equations to row echelon form, then use back substitution to solve the system. 3x + 5y z = 7 x + y + z = 1 x + y + 11z = 7 4) Find det 7 5 4 5) Find 3 1 det 1 4 1 i j k 6) Find det 3 1 4 1
p MAC1140 Pre-Calculus Algebra Sample Test 4 Falzone page 3 of 3 Find the Partial Fraction Decomposition for the following rational expressions: 7) 5x 3 ( x + 1 ) ( x 3 ) 8) 3 4x + x + x + x ( x + 1 )
tl T.7 : :::.:..:... : l:!.i:.: ' 1: fi,:.,.,t.:.,.- t,r.:,,. 1::::: ::::: ffi'qwtrafl ',""''"' '' :':' ':'':': :::'':: '''': "'r i:l,,,,,,,',,,:,,,,' :,!,,:!:!!,!,,.,,::':. 5=..Hil...d...#. i.9..'.'ff).1...@ll. Chapter 8 Figure 7, p. 48 Algebra for College Students, Second Edition, by LiaUMiller/Hornsby Copyright @ 199 by HarpeCollins College Publishers
The Parabola F(0, p) x = p F( p,0) y = p x = 4py y = 4 px p > 0 p > 0 Focus: (0, p ) Focus: ( p,0) Directrix: y = p Directrix: x= p Eccentricity = e = 1
The Circle r ( hk, ) ( xy, ) x h + y k = r ( ) ( ) Center: hk, radius: ( ) Eccentricity= e = 0 r
Finding an Equation of a Circle : ( ) ( ) x h + y k = r 1) ) 3) 4) 5) 6)
The Ellipse x a y b + = 1 x b y a + = 1 c = a b c= a b Eccentricity = e e = 0 1 c a < < rounder flatter
The Hyperbola x a y b = 1 y a x b = 1 c = a + b c= a + b c Eccentricity: e=, where e> 1 a As As e 1, graph is narrower e, graph is wider
Eccentricity Eccentricity is a geometric constant, often denoted e, that describes the shape of a conic section. The eccentricity of a conic section is the ratio of the distance of any point on the curve from a given fixed point (the Focus), to the distance of that point from a given fixed line (the Directrix). This constant is independent of the position, orientation, and size of the curve, and so identifies a family of similarly shaped curves. e = distance from any point on the curve to its focus distance from that point on the curve to its directrix
ECCENTRICITY Circle Ellipse Parabola Hyperbola 0 1
Find an Equation of the Following Conic Sections 1) ) 3) 4) 5) 6)
Analyze the Following Conic Sections 1) ) Equations Vertex Form: Standard Form: Domain: Range: Vertex: Focus: Eccentricity:
Analyze the Following Conic Sections 3) 4) Equation Standard Form: Domain: Range: Vertices: Foci: Eccentricity:
Analyze the Following Conic Sections 5) 6) Equation Standard Form: Domain: Range: Vertices: Asymptotes: Foci: Eccentricity:
Conic Sections Summary Equation Graph Description Focus/Foci Eccentricity Circle Parabola Parabola Ellipse Ellipse
Conic Sections Summary Equation Graph Description Foci Eccentricity Hyperbola Hyperbola
MAC1140 Test 3 Sample Name Find an equation of the following: 1) ) 3) 4) 5) 6)
p MAC1140 Pre-Calculus Algebra Test 3 Sample Falzone page of 4 7) Graph y = x + 8x + 3 B) Write in Vertex form. C) Find the coordinates of the focus. Graph the focus. 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 - -3-4 -5-6 -7-8 y 8 x 8) Write x + y + 8x 6y + 16 = 0 in standard form, then graph it. -8-7 -6-5 -4-3 - -1-1 1 3 4 5 6 7 8 - -3-4 -5-6 -7-8 y 8 7 6 5 4 3 1 x
p MAC1140 Pre-Calculus Algebra Test 3 Sample Falzone page 3 of 4 Analyze the Following Conic Sections 9) 10) Graph: ( x ) ( y 4) + = 1 Graph: 9 4 ( x + 6) ( y 3) = 9 16 1 Domain: Domain: Range: Range: Vertices: Vertices: Foci: Foci: Eccentricity: Eccentricity:
p MAC1140 Pre-Calculus Algebra Test 3 Sample Falzone page 4 of 4 Analyze the Following Conic Sections 11) 1) Graph: x ( y 3) 4 x + = ( y 1) 3 = + + Graph: ( ) 1 Domain: Range: Vertex: Domain: Range: Vertex: Focus: Focus: Eccentricity: Eccentricity:
Taylor Series Expansions of Functions 1 1 x = + + + + + 3 4 1 x x x x... for x ( 1,1) e x 3 4 x x x x = 1 + + + + +... 1!! 3! 4! 3 4 ( x 1) ( x 1) ( x 1) ln( x) = ( x 1) + +... 3 4 3 5 7 9 x x x x sin( x) = x + +... 3! 5! 7! 9! 4 6 8 x x x x cos( x) = 1 + +...! 4! 6! 8! x x x x = + + for x [ 1,1] 3 5 7 9 3 5 7 9 1 tan x x...
Doubling Rice Grains??? 1 4 8 16 1. How many grains of rice will there be on the square b?. How many grains of rice will there be on the square h? 3. How many grains of rice will there be on the square h8? 4. How many total grains of rice will be on the board after placing rice on square b? 5. How many total grains of rice will be on the board after placing rice on square h? 6. How many total grains of rice will be on the board after placing rice on square h8?
The Binomial Theorem For any positive integer n, n n n n ( x + y) = x + x y + x y + x y + + x y 1 3 r 1 n n n 1 n n 3 3 n ( r 1) r 1 n n + + x y + xy + y n n 1 n n 1 n rth term Or, alternately, n n n 0 n n 1 1 n n n n 3 3 n n ( r 1) r 1 ( x + y) = x y x y x y x y x y 0 + 1 + + + + 3 r 1 n n n + + x y + x y + x y n n 1 n n 1 n 1 0 n rth term where n n! ncr r = = r!( n r)! Using Summation Notation, n n n n n r r n r r ( x+ y) = x y = ncr x y r= 0 r r= 0
The Binomial Theorem For any positive integer n, n n n n 1 n n n n 3 3 n n ( r 1) r 1 ( x y) x x y x y x y x y 1 3 r 1 n n x y x y y n n 1 n n 1 n rth term Or, alternately, n n n 0 n n 1 1 n n n n 3 3 n n ( r 1) r 1 ( x y) x y x y x y x y x y 0 1 3 r 1 n n n x y x y x y n n 1 n n 1 n 1 0 n rth term Using Summation Notation, n n n n r r n r r ( x y) x y ncr x y r 0 r r 0 n where n n! ncr r r!( n r)! 1) Use the Binomial Theorem and expand x y 5. ) Use the Binomial Theorem and expand x 3 6. 3) Use the Binomial Theorem and find the 7 th term of x 10.
Principle of Mathematical Induction Mathematical Induction is a method of mathematical proof typically used to establish that a given statement is true for all positive integers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. The simplest and most common form of mathematical induction proves that a statement involving a positive integer n holds for all values of n. The proof consists of two steps: 1. The basis (base case). Show that the statement is true for n = 1.. The inductive step. Show that: If the statement is true for n = k, then the statement is also true for n = k + 1, where k is a positive integer. k {1,,3,4,...} Another way of stating step. is: Prove that the statement is true for n = k + 1, assuming that the statement is true for n = k.
MAC1140 Sample Final Exam Name 1) Graph f ( x) x + 3 for 4 x < 1 = x + x < ( ) 4 for 1 5 ) Find the Difference Quotient f ( x + h) f( x) h of the function f ( x) = x + 3x + 3) Refer to the graph at the right. A) Find a piecewise-defined function of the graph. f ( x) = Find the following. Refer to the function above. B) lim f( x) x 4 C) lim f( x) x 4+ D) lim f( x) x 4 E) f ( 4) F) f ( ) G) lim f () x x 1 H) lim f( x) x + I) lim f( x) x J) f () 4) Expand the following logarithms to the Sum or Difference of Logarithms. Simplify Completely. log w y x 3 4 z 5) Condense the following logarithmic expressions to a single logarithm. Simplify Completely. 1 3 log x log y 4log z
p MAC1140 Pre-Calculus Algebra Sample Final Exam Falzone page of 6 Sketch the following functions, showing all asymptotes and intercepts. Give equations of all asymptotes. Show your work. 6) f ( x) = 3x 4 x + Find the following Limits: A) B) x + lim f( x) lim f( x) x C) lim f( x) x + 7) The formula 0 0.3 t -8-7 -6-5 -4-3 - -1-1 1 3 4 5 6 7 8 - D = D e can be used to find the number of milligrams (mg) D of a certain drug that is in a patient s bloodstream t hours after the drug has been administered where D 0 is the initial amount of the drug administered. Assume that 10 mg of the drug is administered initially. A) Use your calculator to determine how much of the drug is in the patient s bloodstream 3 hours after it was administered. Round your answer to 1 decimal place. -3-4 -5-6 -7-8 y 8 7 6 5 4 3 1 x B) When the amount of the drug that is in a patient s bloodstream reaches 5 mg, the drug needs to be readministered. How long until the drug needs to be re-administered? Solve by hand. Show your work. Round your answer to 1 decimal place. Solve the following equations algebraically, then use your calculator to round your answers to decimal places. 8) ln(x 5) = 3 9) x + 4 = 6 10) Solve A = P 1 + r n nt for t
p MAC1140 Pre-Calculus Algebra Sample Final Exam Falzone page 3 of 6 Find an equation of the following: 11) 1) 13) 14) 15) Write x + y + 8x 6y + 16 = 0 in standard form, then graph it. 7 6 5 4 3 1 x -8-7 -6-5 -4-3 - -1 1 3 4 5 6 7 8-1 - -3-4 -5-6 -7-8 y 8
p MAC1140 Pre-Calculus Algebra Sample Final Exam Falzone page 4 of 6 16) Use your calculator to solve the following system of linear equations. x 3y + z = x + y z = 10 5x 3y + 4z = 5 i j k 17) Find det 3 1 4 1 18) SETUP the Partial Fraction Decomposition of the following rational expressions. Do NOT solve the resulting equations; do NOT find the values of A, B, etc. 5x + 3x 4 3 x ( x + 4) ( x + x + 5) 19) Find the Partial Fraction Decomposition for the following rational expression: 5x 3 ( x + 1 ) ( x 3 )
π MAC1140 Pre-Calculus Algebra Sample Final Exam Falzone page 5 of 6 Arithmetic Sequence: a = a + ( n 1) d 1 n n Arithmetic Series: Sn = a1 + an or Sn = a1 + ( n 1) d Geometric Sequence: a n n = ( ) ( ) n 1 1 n a1(1 r ) a1 a r Geometric Series: Sn = and S = for 1 < r < 1 1 r 1 r 0)A) Let a ( 1) n n n 1 n. Write down the first 5 terms of the sequence. B) Find 4 i 1 (3i 1) C) Given the sequence: 5, 8, 11, 14, 17,, find the 450 th term of the sequence. D) Suppose that the 0 th term of an arithmetic sequence is 98, and that the 1 st term of an arithmetic sequence is 103. Find the first term of that arithmetic sequence. E) Find the sum of the first 10 terms of the sequence: 4, 1, 36, 108,. Solve algebraically, don t just add up the 10 terms. F) Find the infinite sum: 1 1 1 1 1 + + + + + 4 8 16
p MAC1140 Pre-Calculus Algebra Sample Final Exam Falzone page 6 of 6 1)A) Create the first 6 rows of Pascal s Triangle 5 B) Use Pascal s Triangle to expand ( x + y) 4 C) Expand (x + 3) D) Find the 5 th term of the expansion ( x + ) 0 Note: You do NOT need to expand the binomial to find the 5 th term.