3. x - y = 7 yes 2x + 3y = x + 2y = 4. 2x + y = 7 tcs. + y = 7 y = -4x + 7

Transcription

1 ! A. Exercises Is (-2, 5) a solution to each sstem? 1.x + = 3 2x + = 1 2.x 2 = 8 3. x - = 7 es 2x + 3 = x + 2 = 4 n 5x + 3 = 5. S sstem determine whether the given point is the solution. 3x + -= 1 For each 5. x + 3 = 13; (2, -5) 8. 3x - =- -7, (-1, 4) 6> + 7 el x = -19 na 6. 2> - 5 = 9, (7, 1) 9. 5x + 6 = ; (6, -5) x + = 8 vfots 2x + = 7 tcs 7. x + 2 = 1; 3. -2) 1. = -2, (-2, -8) 3> - 2 = 5 rno -= 4x 5x -L- 6 = 2x + = 7 516) + 6(-5) = 2(6) + (-5) = = = 7 = 7 = 7 = - 2 = 4x -8 = -2 False -8 = 4(-2) Ox + = 5 = -x = -8 x - = 1 - = -x + 1 = x - 1 B. Exercises Graph each sstem of linear equations to find its solution. 11. x = x + = 7 x - = 1 '3 2' x x - = x + = x > + = x - = -5 = 1 12x ) 14. x + 2 = x - 5 = = -6 2x - = i - 3 = , -4) x - = -2 - = -x - 2 = x + 2 4x - = 4 -= -4x + 4 = 4x - 4 C. Exercises Graph the following equations and estimate the solution of the sstem. 2. 9x - 2 = 7 3x student answers should approximate the solution (1-,. Currnilati.ve Review Simplif ,; I r ,'* N i 3 ' SOLVING SYSTEMS OF EQUATIONS BY GRAPHING : 1n4.-; x + 2= 1 2= -x 1-1 = x 5 9x + 2 = -6 2 = -9x - 6 = 2 x x - 3 = -12 4x - = -8-3 = -8x = -4x = Tx + 4 = 4x + 8 4x + = 7 = -4x + 7 x - 3 = 18-3 = -x + 18 =x - 6 = 2 61 r I! cot- ii 7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING 275

2 gx-f. - =--x+ 4 nnsistent; Ox = 3 = x+ 3 = x 3 5x + = 8 5x + 8 4x = 15 = 4x + 15 = 4x 15 Definitions Since the slope-intercept form of these two equations is the same, their graphs are identical. The line described b the equation = x -1-3 is the graph for both equations. Thus. the entire line is the solution to the s stem. The line contains an infinite number of points all of which are solutions. An linear sstem of equations that produces an infinite number of solutions is called consistent but dependent (not ). To summarize, a sstem of two linear equations ma have. 1, or an infinite number of solutions. Use the slope-intercept form of the equations to determine the number of solutions. A consistent sstem of equations is a sstem that has at least one solution. A dependent sstem of equations is a consistent sstem that has an infinite number of solutions. Notice that inconsistent means not consistent and means not dependent. POSSIBLE SOLUTIONS FOR SYSTEMS OF EQUATIONS Independent Consistent Dependent Inconsistent Number of solutions finite infinite none Graphs graphs graphs graphs do not intersect coincide intersect Applied to lines (number of solutions) one all points on the line none + = 8 2x + 8 c ependent graph marked off b 2's 4x + 2 = 16 2= 4x + 16 = 2x + 8 A. Exercises Solve each sstem b graphing and tell whether it is consistent or inconsistent. If the sstem is consistent, tell whether it is dependent or. 1. x + = 4 3); consistent: 5. x = 4); consisten',. 5x + = 8 3x + 2 = x = 3 (4. 1); consistent, 6. 8x 2 = 6 no solution; 4x = 15 4x + 3 inconsistent 3. 2x + = 8 :,ntire line; consistent, 7. x 2 = 4 1.7,, 3); consistent, 4x + 2 = 16 dependent 3x + 2 = x + = 5 i".;; 5); consistent, 8. x + 2 = 8 ern:pie line; consistent, 4x + 3 = 15 = 1+ 4 dependent 278 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES 3x -= 5 1 3x + 5 4x + 3 = 15 3 = 4x + 15 = 3 4 x + 5 Case 3 1. Lines coincide. 2. Solution is an infinite set. 3. Sstem is consistent and dependent. Example x + = 3 2x + 2 = 6 As ou discuss this case, point out that the slopes and -intercepts of the two equations are the same and that the equations are equivalent. Ultimatel, the equations for a dependent sstem are the same and the solution is the entire line. This information is summarized for the students in the table on page 278. (In nonlinear dependent sstems. the solutions of one equation are a subset of another.) Have the students solve the sstems in examples 1, 2, and 3 b graphing the equations and, identif the tpe of sstem. Common Student Error. Students ma have difficult with the terminolog in this section. Speak carefull and accuratel as a model for them. n 278 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

3 $ B. Exerciies Solve each sstem b graphing and tell whether it is consistent or inconsistent. If the sstem is consistent, tell whether it is dependent or. 9. x= x + 4 = 4 - x= 4.'7" x 2 = 8..ant 1. x = 6 _ 16. 3x + = 5( ;iista t, - C. Exercises 5 = 3.dependent = 3x x 3 = 12 consis;:er.,.., 17. 3x 5 = 15 :15, 5); consistent, 2x + 3 = 9 ':?de p endent 3x + 5 = Zr 7 = x + = 4 ;3, 7); consistent, 3x + 7 = 14 :nuep2ndent = 7 incịependent 13. 3x + 4 = 2 2); consistent, 19. x 3 = 15 4); consistent, x + 2 = 8 : i?dependent = 411,--iependervi 14. x = 4.3, 7); consistent, 2. 12x + 4 = 8 no solution; x + 3 = 24 3 = 15 9x inconsistent Draw (onclusions about the slopes and -intercepts of the following tpes of sstems of linear equations. 21. inconsistent sstems The 1ineL: have the same slope but different -intercepts. 22. dependent sstems The lines have the same slope and the same -intercept. 23. sstems The iines "nava different ;lopes. <a t^aiatir^e Review Solve. 24. x 5 = 8 x = 13!4.1] 25. 5x 11 = 2x + 4 x = 3 [4.3] = [4. 7] 27. 4(x 3) 8x i( 3 [5.4] 28. 2x + 1 > 19 x > 9 or x < 1 [5.7] x = 3x 2 = 14 2 = -x = -3x = + 5 8x - 2 = 6-2 = -8x + 6 = 4x - 3 = 2 3 x + 7 = 4x + 3 x 2 = 4 3x + 2 = 12 2= x+ 4 2=-3x-12 = -Tx 2 = 2 x 6 X 7.2 USING THE GRAPHING METHOD 279 x 2 = 3 2 = -x -- 8 = 2 1 x + 4 = 21 x + 4 (tx = 3 incons'stent x = 4 3 x_ = 6 - = -x + 6 = x - 6 5x 5 = 3 5 = 5x + 3 = x USING THE GRAPHING METHOD 279

4 Answers 2x 7 = 21 3x + 7= 14-7= -2x 21 7= -3x = -Tx - 3 = - 73 x + 2 Chapter 7 Sstems of Equations and Inequalities 11x + = 19 = -11x x - 2 =7 3x - 4 = x x =- x -- T 3x + 4 = 2 4 = -3x x + = 8 2= -x x + 4 t 9x - = -5 - = -9x - 5 Y - 9x 5-5 = -3 2x - = 3-5 = -x - 3 = 2x = -5-x =2x x - = -8 -=-12x- 8 12x + 8 = ± _ 37 5 ' ' _= = 2 3 = 2 i T 6_2 2_1 = g \ = /527 2 _ /529 _ /23 2 _ 23 V X + 1 5x - 3 = -12 2x + 3 = -9-3 = -5x = 2x Tx + 4 = x x x - = -4 x + 3 = 24 -= -x = -x + 24 = x + 4 = x = 4 x - 2= 8 4 = -3x + 4-2= -x = T x - 4 ANSWERS 67

5 3x + = 5-3x + 5 = 3x x + 4 = 8 3 = 15-9x 4 = -12x + 8 = 5-3x = -3x + 2 = -3x + 5 3x 5= 8 3x = X = x -- 5 = -15 3x -+ 5 = 45-5 = -3x = -3x x x + 9 x + = - 4 = -x - 4 x - 3 = -15 = 4-3 = -x - 15 = -Tx 5 (-3, 4) v~ -7 x A- P 5x x + 4 3x = 15 x = 5 x ( 32 1 x + 4)= 5x + 12 = 7 5x = 58 «= (x 4-3) s 8x 4x - 12 _s_ 8x -4x _s 12 9 x -3 5 (-6-)16 2x + >19 2x + 1 > 19 or 2x + 1 < -19 2x> 18 2x< -2 x > 9 x < -1 3x + = -2 = -3x - 2 x - 5( " 23 = 6 23x + 75 = x~ x = 23 2x - 8 = 7 2x - 8( -3x - 2) = 7 2x + 24x + 16 = 7 26x = -9-9 ^- 26 3(4) += Y = = i ) Y = 26 \ 26, 26 / x - 5 = 6 4x + 3 = 9 x = (5 + 6) + 3= = 9 23= Ans. ( -2./35 ) 2 = x + 9 4x + 2 = 29 4x + (x + 9) = 29 5x = 2 x = \i2 = V49 = An (4, 7), (4, -7) 2x - 4 = 7 2(-3- - ) - 4 = ~ = 5-5 = 22 3: _± 5 ( -2!): 8 66x - 25 = x = _ = 25x 3 = 25x , _ = 3^ Y = -6- x 1v- 12 = , 1x - 12( -Tx - = x - 1x = 14 True Ans. entire line 32 ± 9 = 2x 32 = 2x- 9 _ 2 9 Y - 32 X 32 _ 5 Y - ^ x - 24 = 7 15x u ^ 392 ) = 7 15x - 15x + = 7 27 = 7 False Ans. no solution (1 1) 7. x x x 2 Ans. ( ) x 2 < 6 2 < - x < 2 X ± 3 68 ANSWERS

6 Write two equations and solve the sstem b substitution. x = 63 x = The sum of two numbers is 63. and their difference is 13. Find the two = 63 x x (63 x) = 13 numbers. :=E x 63 + x = The sum of two numbers is 996. The difference of the larger and twice 2x = 76 the smaller is 33. Find the two numbers. - x = = 63 C Exercises = 25 Ans. (38, 25) Solve each sstem b the substitution method. 19. (x ) 2 = x2 + = 25 x + = 6 e ies 2X2 32 = 5-2. x2 = 5 8x = (1, 4), (-1 ' e Review Graph. Use number lines or Cartesian planes as appropriate (5, 1), (-1, 2)) 23. x 5 = 7 r x > x I = 2 I./.6] 26. = 5x > 2 x < 1 1 A Pi 2. 2 L I. 2. x + = 996 x 2 = 33 = 996 x x 2(996 x) = 33 x x = 33 3x = 225 x = = 996 = 321 x + = 6 Ans. (675, 321) (X ± ) 2 = 36 (6) 2 = = 36 True Ans. entire line 2+ = 5 8x2 2 = = 5 x 2 8x2 2(5 x 2) = 8x x2 = 1x2 = 1 X2 = 1 x = -± = 5 (-1)2 = = = 5 = 4 = 4 Ans. (1, 4). ( 1, 4) x = 25 2x2 3 2 = 5 2 = 25 x 2 2x2 3(25 x 2) = 5 2x x 2 = 5 5x 2 = 8 x2 = 16 x = = 25 (-4)2 + 2 = = = Y = = -±3 Ans. (4, 3), (4, 3). (- 4, 3), (- 4, 3) 7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD 287 [6.1] (-1 '2) 5,1) x x < 2 [5.2] 4 I 1 I I m I I> [6.6] C ) [6.1] 7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD 287