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05x -L- 6y = 0 • A. Exercises Is (-2, 5) a solution to each system? 3. x - y = 7 2x + 3y = 4 1. x + y = 3 2x + y = 1 yeS 2. x 2y = 8 3x + y -= 1 n 0 00 4. 3x + 2y = 4 €. S 5x + 3y = 5 For each system determine whether the given point is the solution. 8. 3x - y =- -7, (-1, 4) 5. x + 3y = 13; (2, -5) x - 5y --= -19 na 6> + y 7 el 9. 5x + 6y = 0; (6, -5) 6. 2> - 5y = 9, (7, 1) 2x + y = 7 tcs x + y = 8 vfotS 10. y = -2, (-2, -8) 3. -2) 1; 7. x + 2y = y -= 4x 3> - 2y = 5 rNO 516) + 6(-5) = 30 - 30 = 0 0 = 0 y = - 2 -8 = -2 False Ox + y = 5 y = -x + 5 2x + y = 7 0 2(6) + (-5) = 7 12- 5=7 7=7 y = 4x -8 = 4(-2) --8 = -8 x-y=1 -y = -x + 1 y=x- 1 B. Exercises Graph each system of linear equations to find its solution. 16. 4x + y = 7 11. x y = 5 x - 3y 18 13 . x - y = 1 '3 2' 17. x + y = 9 12. x - y = -2 11x + y 19 - y 4 18. 9x - y = -5 13. 6> + y = 7 12x - y -8 y=1 19. x - 5y = -30 14. x + 2y = 10 2x - y = 3 9 y + 2y= -6 0 -4) x - y = -2 -y = -x - 2 y=x+ 2 4x - y = 4 -y= -4x + 4 y = 4x - 4 15. 8i - 3y = -12 y - - 8 -3, -4) C. Exercises Graph the following equations and estimate the solution of the system. 20. 9x - 2y = 7 14 student answers should approximate the solution (1-,. 3x + 4y Currnilati.ve Review 3. Simplify. 3 12 49 21. 5 + 35 • 9 37 15 ,;12.31 24. 6-2 • 2-i 14 1 39 -3 ,'*-1 25. 22. I r 266 1 8 • N 343 9- 3- 3 527 1 144 + 72 3 ' 31 - 7 275 7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING :WWW.F7•, x + 2y= 10 2y= -x 10 -1 = x 5 01n4 .-; 9x + 2y = -6 2y = -9x - 6 y= 2 x - 3 = 2 61 8 8x - 3y = -12 4x - y = -8 -3y = -8x - 12 -y = -4x - 8 y = Tx + 4 r I ! cot! y = 4x + 8 8 0 4x + y = 7 y = -4x + 7 3y = 18 -3y = -x + 18 y =x - 6 x - • ii 7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING 275 Since the slope-intercept form of these two equations is the same, their graphs are identical. The line described by the equation y = —x -1- 3 is the graph for both equations. Thus. the entire line is the solution to the s y stem. The line contains an infinite number of points all of which are solutions. 5x + y = 8 — 5x + 8 gx-Fy.-- y =--x+ 4 nnsistent; independent y Any linear system of equations that produces an infinite number of solutions is called consistent but dependent (not independent). To summarize, a system of two linear equations may have 0. 1, or an infinite number of solutions. Use the slope-intercept form of the equations to determine the number of solutions. Definitions A consistent system of equations is a system that has at least one solution. A dependent system of equations is a consistent system that has an infinite number of solutions. Ox — y = 3 —y =— x+ 3 y=x — 3 4x — y = 15 —y= —4x + 15 y = 4x — 15 Notice that inconsistent means not consistent and independent means not dependent. POSSIBLE SOLUTIONS FOR SYSTEMS OF EQUATIONS Consistent Inconsistent Independent Dependent finite infinite none graphs intersect graphs coincide graphs do not intersect one all points on the line none Number of solutions Graphs Applied to lines (number of solutions) ► A. Exercises graph marked off by 2's ®+y=8 y —2x + 8 Solve each system by graphing and tell whether it is consistent or inconsistent. If the system is consistent, tell whether it is dependent or independent. 4); consisten',. 1. x + y = 4 3); consistent: 5. x + 2y — 10 = 0 3x + 2y = 14 independent 5x + y = 8 independent 2. x — y = 3 (4. 1); consistent, 6. 8x — 2y = 6 no solution; 4x — y = 15 independent y 4x + 3 inconsistent 3. 2x + y = 8 :,ntire line; consistent, 7. x — 2y = 4 1—.7,, —3); consistent, 4x + 2y = 16 dependent 3x + 2y = —12 independent 4. 3x + y = 5 i." ;; 5); consistent, 8. x + 2y = 8 ern:Pie line; consistent, y = —1+ 4 dependent 4x + 3y = 15 independent 4x + 2y = 16 2y= —4x + 16 y = —2x + 8 consistent; c ependent y 278 01 ► 3x -y= 5 y —3x + 5 4x + 3y = 15 3y = —4x + 15 —4 y= 3 x + 5 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES Case 3 1. Lines coincide. 2. Solution is an infinite set. 3. System is consistent and dependent. Example x+y=3 2x + 2y = 6 As you discuss this case, point out that the slopes and y-intercepts of the two equations are the same and that the equations are equiv- alent. Ultimately, the equations for a dependent system 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 systems. the solutions of one equation are a subset of another.) Have the students solve the systems in examples 1, 2, and 3 by graphing the equations and, identify the type of system. Common Student Error. Students may have difficulty with the terminology in this section. Speak carefully and accurately as a model for them. n 278 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES $•• B. Exerciies x + 2y - 10 = 0 3x 2y = 14 2y = -3x + 14 2y = -x + 10 Solve each system by graphing and tell whether it is consistent or inconsistent. If the system is consistent, tell whether it is dependent or independent. 9. x= —3 x= 4 .'7" 10. x — y = 6 15. 3x + 4y = 4 x — 2y = 8 16. 3x + y = _ 1 5( y= = + 5 consistent; independent ..ant 3 2 x + 7 y ;iista t, — 5y = 30 .dependent y = 3x — 4 independent consis;:er.,.., 17. 3x — 5y = —15 :15, 5); consistent, 11. 5x — 3y = —12 2x + 3y = — 9 ':?de p endent 12. Zr — 7y = 21 3x + 7y = 14 :nuep2ndent 2); consistent, 13. 3x + 4y = 20 x + 2y = 8 : i?dependent 14. x — y = — 4 .3, 7); consistent, x + 3y = 24 independent 3x + 5y = 45 independent 18. x + y = —4 ;3, —7); consistent, y = — 7 inci.ependent 19. x — 3y = — 15 4); consistent, y = 411,--iependervi 20. 12x + 4y = 8 no solution; 3y = 15 — 9x inconsistent 8x - 2y = 6 -2y = -8x + y = 4x - 3 0- C. Exercises Draw (onclusions about the slopes and y-intercepts of the following types of systems of linear equations. y= 4x + 3 6 21. inconsistent systems The 1ineL: have the same slope but different y-intercepts. 22. dependent systems The lines have the same slope and the same y-intercept. 23. independent systems The iines "nava different ;lopes. X <a t^aiatir^e Review Solve. 24. x — 5 = 8 x = 13 !4.1] 25. 5x— 11 = 2x + 4 x = 3 [4.3] 26. 3. + 40 = 176 27. 4(x — 3) 28. 8x i( 8 [4. 7] x — 2y = 4 3x + 2y = —12 2y= —2y= —x+ 4 2y=-3x-12 —3 [5.4] y= 2x + 1 > 19 x > 9 or x < — 10 [5.7] y= -T x — 2 2 x 6 consistent; independent y 7.2 USING THE GRAPHING METHOD 3 2y = -x -- y= 21 x 2y = 8 1 y= 2 x + 4 x+4 (tx = —3 x =4 incons'stent y 279 3 x_ y = 6 -y = -x + y=x-6 6 5x — 5y = 30 —5y = —5x + 30 y=x-6 7.2 USING THE GRAPHING METHOD 279 0 Answers 2x 7y = 21 3x + 7y= 14 -7y= -2x 21 7y= -3x + 14 2 y = -Tx - 3 y= 73 x + 2 y Chapter 7—Systems of Equations and Inequalities 9x - 2y =7 3x - 4y = 14 0 -2y- -9x + 7 4y - -3x + 14 9 7 -3 7 y=- x -- T 11x + y = 19 y= -11x + 19 0 3x + 4y = 20 4y = -3x + 20 - +5 consistent; independent t • • y 0 9x - y = -5 -y = -9x - 5 Y - 9x 5 12x - y= -8 -y=-12x- 8 12x + 8 y= 0 4 7 3 ± 12 49 5 35.9 5 3 3 28 9 28 _ 37 5 ' 15 - 15 ' 15 - 15 7 14 28 21 26 39 _= 78 78 y x+ = 8 2y= -x + 8 -1 y - 2 x+4 x - y = -4 -y= -x - 4 -7 78 7 y= x + 4 x + 3y = 24 3y = -x + 24 -1 y= 3 x + 8 3x + 4y = 4 4y = -3x + 4 x - 2y= 8 -2y= -x + 8 78 8 = 23 = 2 i 343 T 73 6_2 2_1 = 1 3 g-3 62.3 333 1 9 • 9 -9 663 2 221 0 93 2 27 8 527 1 = \ 144 72 +/527 2 _ 144 + 144 - - 5y = -30 -5y = -x - 30 1 y= -5-x 4 y 2x - y= 3 y= 2x + 3 /529 _ /23 2 _ 23 V 144 - 12 212 y =2x - 3 0 -3 X + 1 5x - 3y = -12 2x + 3y = -9 -3y = -5x - 12 3y = 2x 9 5 -2 y - -Tx + 4 y= 3 x - 3 y = Tx - 4 consistent; independent y •• consistent; independent y ANSWERS 607 0 3x + y = 5 y -3x + 0 y = 3x - 4 5 y 12x + 4y = 8 4y = -12x + 8 3x 5y= 8 3x = 8 - 5y 3y = 15 - 9x y = 5 - 3x y = -3x + 5 y = -3x + 2 8 5 X = - -3-y 2x - 4= 7 2(-3- - ÷y) - 4y = 16 - - 4y __ y AP x 7 7 16 - 10y- 12y~ 21 -22y = 5 -5 y= 22 2!): 3 : _± 5 ( - 3x -- 5y = -15 3x -+ 5y = 45 -5y = -3x - 15 5y = -3x + 45 3 y - -53 x + 9 y 5 x + 3 consistent; independent 0 8 5x - 11 3x = 15 x=5 2x + 4 66x - 25 = 176 66x = 201 _ 67 66 - 22 201 176 160( 321 x + 40)= (-6-)160 312 x - 2535 , _ 30 30 y= 30^ 7 5 Y = -6- x «= 5 0(x 4 - 3) s 8x 4x - 12 _s_ 8x -4x _s 12 x -3 2x + >19 2x + 1 > 19 or 2x> 18 x > 9 9 0 -7 0 3x + 10v- 12y = 14 7, 2 5 = 10x - 12(-Tx - 7 2x + 1 < -19 2x< -20 x < -10 2x - 8y = 7 2x - 8( -3x - 2) = 7 2x + 24x + 16 = 7 26x = -9 y = -2 y = -3x - 2 -9 ^- 26 -2 2 26 Y -27 + 26y = -52 26y = -25 0 i -25 Y = 26 x x= -9 -25 ) \ 26 , 26 / 3y = 9 4(5y + 6) + 3y= 9 20y + 24 + 3y = 9 23y= -15 5y = 6 5y + 6 4x + consistent; independent -15 x- 5( 2 135 - 23 6 „ 75 " 23 = 6 (-3, 4) 75 = 138 23x~ 63 23x + v~ x = 63 23 y2 = x + Ans. ( 9 -2./35 ) 4x + y2 = 209 4x + (x + 9) = 209 5x = 200 x = 40 40 + 9 V49 \iy2 = y= 608 ANSWERS 10x - 10x + 14 14 14 = 14 True Ans. entire line 32y ± 9 = 20x 32y = 20x- 9 _ 20 9 Y - 32 X 32 _5„ ^ - 32 9 Y- 15x - 24y = 7 15x - 241 u ^ 15x - 15x + 392 ) = 7 =7 = 7 False Ans. no solution -27 y=4 14 1 27 3 (4) +y= x - 3y = -15 -3y = -x - 15 y = -Tx 5 ) Ans. ( 35 + 30y= 25x 30y = 25x - 35 5x + 12 = 70 5x = 58 x + y = -4 y = -x - 4 8 An (40, 7), (40, -7) (1•10) 7. x + 2y -4 2y -x - 4 -1 2 x 2 y 2y < 6 2y < - x + x y< 6 2X±3 -1 x y = 63 y = 63 — x Write two equations and solve the system by substitution. 17. The sum of two numbers is 63. and their difference is 13. Find the two numbers. :=E 18. The sum of two numbers is 996. The difference of the larger and twice the smaller is 33. Find the two numbers. C Exercises Solve each system by the substitution method. 21. x2 + 19. (x y)2 = 36 e iEs 2X2 x + y = 6 20. x2 y = 5 8x2 -- 2y = 0 (1, 4), (-1 ' ye 0 = 25 3y2 = 5- Review < 6 A‘Pi 1 1 — 2. Ans. (38, 25) x — 2y = 33 x + y = 996 y = 996 — x x — 2(996 — x) = 33 x — 1992 + 2x = 33 3x = 2025 x = 675 675 + y = 996 Ans. (675, 321) y = 321 x+y= Graph. Use number lines or Cartesian planes as appropriate. 25. x I = 2 I./.6] 22. 1(5, 1), (-1, 2)) 26. y = —5x + 3 23. x 5 = 7 r4.11 0 2 L+ (X ± y)2 = 36 (6)2 = 36 36 = 36 True Ans. entire line 27. y> 2 — x 24. 5 -- x > 3 38 + y = 63 y = 25 x — y = 13 x — (63 — x) = 13 x — 63 + x = 13 2x = 76 x = 38 • 10 4 8x2 — 2y = 0 2+ y = 5 y = 5 — x 2 8x2 — 2(5 — x2 ) = 0 8x2 — 10 + 2x2 = 0 10x2 = 10 X2 = 1 x = -± 1 -I. 0 2. (-1)2 y = 1 2 + y = 5 1+y=5 1 + y = 5 y=4 y = 4 Ans. (1, 4). ( —1, 4) 5 x2 + y2 = 25 2 y2 = 25 — x 2x2 — 3y2 = 5 2x2 — 3(25 — x2 ) = 5 2x2 — 75 + 3x 2 = 5 5x2 = 80 x2 = 16 x= 42 + y 2 = 25 16 y2 = 25 (-4)2 + y2 = 25 16 + y2 = 25 y2 y2 9 9 y = -±3 Y = Ans. (4, 3), (4, —3). ( - 4, 3), ( - 4, — 3) 287 7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD [6.1] x < 2 [5.2] 4 y I -2 1 I 0 I m 2 I I> ) C [6.10] 4 0 [6.6] (-1 '2) 5,1) • x 7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD 287