# Algebra 2. Systems of Equations Unit 3. Name:

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1 Algebra 2 Systems of Equations Unit 3 Name: 1

2 Notes Section 4.1 Date Graphing Systems By the end of this section, you should be able to: - Determine by graphing if a system of equations has a unique solution, no solution or an infinite number of solutions! 1. Systems of equations Two or more equations that have the same variables, such as 2x y= 10 x+ 3y= 9 form what is called a system of equations. System Solution Set When graphing a system of 2 linear equations in 2 unknowns there are exactly three possibilities as illustrated by the examples that follow. Example 1: Graph the following equations: y = x y = x + 2 The solution of this system is: The solution is and 2

3 Example 2: Soln : This system is and. Soln : This system is. Example 3: Graph the following equations: y = 3x 1 y = 3x 5 The solution of this system is: The solution is 3

4 Example 4: Graph the following equations: y = 3x + 2 6x 2y = 4 The solution of this system is: The solution is and consistent dependent independent inconsistent Homework Sec

5 Homework Sec.4.1 Solve each system by graphing. y= 2x 10 x+ y = x+ 2y = 1 y= x x+ 2y = 1 x 2y = 1 4. x= y+ 1 y = x 1 Answers: (5, -3) (3, -4) Ø 5

6 Classify each system graphed below as one of the following: a) consistent & dependent b) consistent & independent c) inconsistent DO NOT GRAPH! Simply compare the slopes (and if necessary, the y-intercepts) and then classify each system as one of the following: a) consistent & dependent b) consistent & independent c) inconsistent 9. 3 y = x y = x y = 2x+ 7 y = x x+ 4 y = 3 4x+ 8y = x+ 4y = 2 6x+ 4y = 4 6

7 Notes Section 4.2 Solving systems of equations by substitutions Date By the end of this section, you should be able to solve a system of two equations in two unknowns using substitution. Review: 1) Find the slope of the line containing (2, 4) and ( 3, 7). 2) Find the y-intercept: y = 3x 4. 3) Find the x-intercept: y = 3x 4. 4) Simplify: 32 3(3 + 4x) 5) Write an imaginary number, an irrational number, an integer. Example 1: Solve the following system: y = 3x 2x 4y = 10 Example 2: Solve the following system: x = 4y 2x 4y = 8 Example 3: Solve the following system: x + 2y = 2 5x 3y = 2 7

8 Homework Sec. 4.2_(day 1) Solve each system using the substitution method. 1. y = 2x 3y+ x= y = 3x+ 1 y 5x= 1 3. y = 3x+ 2 5x 2y = x 3 y = 14 x= y x 3y = 1 x y = x+ 2y = 6 x y = 3 8

9 Tough ones!! These will challenge you. 7. 3x+ 5y = 30 5x+ 3y = x+ 3y = 5 4x+ 7y = x+ 3y = 4 3x 2y = 7 Answers: (2, 4) (0, 1) (3, 11) (16, 6) Ø (5, 3) (1, 1) (1, 2) 9

10 Notes Section 4.2 (day 2) Solving systems of equations by elimination Date By the end of this section, you should be able to solve a system of two equations in two unknowns using elimination. Review: 1) Find the slope: 2x + 3y = 7 2) Square: (3x+ 5 y) 2 3) Simplify: 33 3(3x + 5y) 4) Solve: (2x 5)(x + 1) = 0 5) Solve: x 5 = 7 (remember clean and dirty!) Example 1: Solve the following system: 2x y = 10 x + 3y = 9 10

11 Example 2: Solve the following system: 5x + 2y = 10 5x + 2y = 20 Example 3: Solve the following system: x y = 11 7x + 4y = 22 Example 4: Solve the following system: 2x + 3y = 2 4x 9y = 1 11

12 Homework Sec. 4.2_(day 2) Solve each system using the elimination method. 1. 5x+ 2y = 11 x+ y = 4 2. x y = 11 7x+ 4y = x 7y = 47 2x+ 5y = x+ 3y = 41 6x+ 5y = x+ 7y = 15 2x+ 9y = x 5y = 26 4x 3y = 17 12

13 7. 2x+ 3y = 2 4x 9y = x+ 10y = 24 6x+ 7y = x 7y = 5 10x+ 3y = x 5y = 38 9x+ 2y = 25 These have untidy fractions for answers x 3 y = 11 5x 6y = x+ 4 y = 8 2x 2y = Answers: (1, 3) (-6, 5) (2, -5) (4, 3) (-3, 0) (-1, -7) (, ) (2, -3) (-1, -2) (-3, 1) (, ) (, )

14 Solving systems review Review: 1) Solve: 5x 3x = 17 2) Simplify: 5x 3x 17 3) Solve: (2x + 1)(3x 2 ) = 0 4) Multiply: (2x + 1)(3x 2 ) 5) Solve and state your answer in interval notation: 3x + 7 < 10 Solve each of the following by graphing: 1. y= 2x+ 1 y= x y= x 1 3 y= x+ 4 14

15 Solve by any method: x = x 3y = x+ 2y = 12 6x 2y = x+ 2y = 5 3x 15= 6y 6. x= 2y 2x+ 2y= 4 7. y 3x= 0 2x 3y= x+ y= 10 6x 2y= x+ 4y= 2 5x 7y= x 8y= 47 2x+ 7y= 22 Answers: (2, -3) (3, 1) (4, 9) (2, 1) (4/3, 2/3) (1, 3) (2, 1) (3, -4) 15

16 Notes Section 4.6 Date Solving systems of three equations in three unknowns By the end of this section, you should be able to solve a system of three equations in three unknowns using elimination. Review: 1) Write an equation of the line with no slope passing through (2, 3). 2) Write an equation of the line with slope = 0 passing through (2, 3). 3) Write an equation of the line with slope 2 3 and y-intercept 7. 4) Write an equation of the line with slope 2 3 passing through (2, 3). 5) Write an equation for the line perpendicular to your answer to number 4 passing through (0, 5). Example 1: Solve the following system: 2x y z= 7 3x+ 5y+ z = 10 4x 3y+ 2z= 4 16

17 Example 2: Solve the following system: 2x+ 3y z = 1 x+ 5y+ 3z= 10 3x y 6z= 5 17

18 Homework Sec. 4.6 Solve the following systems: 1. x 2y+ 3z = 3 2x+ y+ 5z = 8 3x y 3z = x+ 2y z = 10 x+ 4y+ 2z = 3 2x+ 3y 5z = x 4y+ 3z = 15 6x+ 2y+ 9z = 13 7x+ 6y 6z = 6 18

19 4. 5x 4y 6z = 21 2x+ 3y+ 4z = 15 3x 7y 5z = x+ 4y = 19 2y+ 3z = 8 4x 5z = 7 Answers: (-4, 1, 3) (1, 2, -3) (2, -1, 1/3) (-1, 1, -5) (3, 5/2, 1) 19

20 Notes Section 4.3 Second-order Determinants Date By the end of this section, you should be able to evaluate a 2 nd order determinant and use Cramer s rule to solve a system of two equations in two unknowns. Review: 1) Solve: 3x + y = 11 and x + y = 5 2) Evaluate ) Evaluate 4 ( 3). 4) Write a rational number. 5) Write an imaginary number. A second-order determinant is a square array of numbers that is evaluated according to the rule: r s ru st t u = Example 1: Find the value of the determinant: 2 3 = 1 5 Example 2: Find the value of the determinant: 2 3 =

21 Cramer s Rule to solve ax+ by= c dx+ ey= f : c b a c f e d f x= y= a b a b d e d e Example 3: Solve the following system: 3x+ 4y= 2 5x 7y= 17 Example 4: Solve the following system: 11x 5 y= 38 9x+ 2y= 25 21

22 Homework Sec. 4.3 Find the value of each determinant: Solve each system using Cramer s Rule: (show the determinants used!!) 7. 5x + 2y = 11 x+ y= x 7 y= 47 2x+ 5y= x+ 7 y= 15 2x+ 9y= x 7y= 5 10x+ 3y= 16 22

23 11. 4x 3y= 11 5x 6y= x+ 4y= 8 2 x 2 y= x+ 5y= 22 2x 10y= x+ 5y= 22 7x 2y= x+ y= 5 6x 3y= 15 Answers: (1, 3) (2, -5) (-3, 0) (-1, -2) (13/3, 19/9) (22/7, -5/14) Ø (139/33, 173/33) 23

24 Notes Section 4.9 Higher-order Determinants Date Be able to use determinants to solve a system of three (or more) linear equations with three (or more) variables. Review: 1) Evaluate 3x 7 when x = 2 2) Evaluate 3x 7 when x = 1 3) Evaluate 4) Evaluate 2 2x x 1 when x= 4 2 2x x 1 whenx= 4 Cramer s Rule with the Calculator Example Use Cramer s Rule to solve the following system: 3x + y = 19 5x 2y = 50 HOW TO SOLVE DETERMINANTS IN THE CALCULATOR 2 nd MATRIX Press the right arrow key so that EDIT is highlighted and press ENTER Enter 2 X 2 for size of Matrix Enter the coefficients for the denominator determinant nd QUIT 2 nd MATRIX right arrow so that MATH is highlighted Press 1 2 nd MATRIX highlight appropriate matrix letter and press ENTER Nx Ex: x = D = y = N y D = Nx: Ny : D: SOLUTION: (, ) 24

25 Homework Sec. 4.9 Use Cramer s Rule and your calculator to solve the following systems: 1) 2x - 3y = -33 4x + 5y = 11 x = D = y = N y D = Nx N x : N y : D: SOLUTION: (, ) ) 4x - 6y = 15-6x + 9y = 11 x = D = y = N y D = Nx N x : N y : D: SOLUTION: (, ) ) 3x + 2y = 30 5x + 4y = 51 x = D = y = N y D = Nx N x : N y : D: SOLUTION: (, ) 25

26 4) x + y = 9 2x - 2y = 7 x = D = y = N y D = Nx N x : N y : D: SOLUTION: (, ) ) 2x y z = 7 3x + 5y + z = -10 4x 3y + 2z = 4 x = D = y = N y D = z = N z D = Nx N x : N y : N z : D: SOLUTION: (,, ) ) 3x + 4y + 2z = 6 x + 3y 5z = -7 5x + 7y 3z = 3 x = D = y = N y D = z = N z D = Nx N x : N y : N z : D: SOLUTION: (,, ) 26

27 7) 3x 2y + 5z = -17 2x + 4y 3z = 29 5x 6y 7z = 7 x = D = y = N y D = z = N z D = Nx N x : N y : N z : D: SOLUTION: (,, ) ) 2x + 2y + 3z = -1 3x 5y 2z = 21 7x + 3y + 5z = 10 x = D = y = N y D = z = N z D = Nx N x : N y : N z : D: SOLUTION: (,, ) ) Careful! Some variables are missing. (Put a 0 for the coefficient of any missing variable) 3x + 4y = 19 2y + 3z = 8 4x 5z = 7 x = D = y = N y D = z = N z D = Nx N x : N y : N z : D: SOLUTION: (,, ) Answers: (-6, 7) Ø (9, 3/2) (25/4, 11/4) (1, 2, -3) (4, -2, 1) (2, 4, -3) (3, -2, -1) (3, 2.5, 1) 27

28 Notes Section 4.10 Systems of Linear Inequalities Date Be able to draw the graph of a system of linear inequalities!! Review: 1) Simplify: 3x 5(2x 7) 2) Simplify: 3x 5x(2x 7) 3) Is 17 an irrational number? What kind of number is it? 4) Write an irrational number. 5) Evaluate 3x 5x(2x 7) when x = 2 Example 1: Graph y> 2x 5 Example 2: Graph the system: y> 2x 5 1 y x

29 Homework Sec Graph the following systems: 1. y < 3x x+ 3 y 6 2x y < x 2y< 10 x+ y < x+ 3y 15 2x+ 6y < 9 29

30 5. y< 3x 4 2x 3y > 6 3 y ( x ) 6. x> 2 y 3 x+ y 4 x+ y< 1 7. y 3x x+ 4y > x+ y 4 2x y > 6 x 2y = 8 (kinda tricky...) 30

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