第11章連立方程式と連立不等式Systems of Equations and Inequalities

Review Exercises復習問題

Systems of Linear Equations: Two Variables連立一次方程式: 二変数

For the following exercises, determine whether the ordered pair is a solution to the system of equations.

次の各問について、その順序対がこの連立方程式の解かどうかを判定せよ。

1. $\begin{array}{l} 3 x - y = 4 \\ x + 4 y = - 3 \end{array}$ and $(- 1, 1)$

1. $\begin{array}{l} 3 x - y = 4 \\ x + 4 y = - 3 \end{array}$ と $(- 1, 1)$

2. $\begin{array}{l} 6 x - 2 y = 24 \\ - 3 x + 3 y = 18 \end{array}$ and $(9, 15)$

2. $\begin{array}{l} 6 x - 2 y = 24 \\ - 3 x + 3 y = 18 \end{array}$ と $(9, 15)$

For the following exercises, use substitution to solve the system of equations.

次の各問について、代入を使ってこの連立方程式を解け。

3. $\begin{array}{l} 10 x + 5 y = −5 \\ 3 x - 2 y = −12 \end{array}$

4. $\begin{array}{l} \frac{4}{7} x + \frac{1}{5} y = \frac{43}{70} \\ \frac{5}{6} x - \frac{1}{3} y = - \frac{2}{3} \end{array}$

5. $\begin{array}{l} 5 x + 6 y = 14 \\ 4 x + 8 y = 8 \end{array}$

For the following exercises, use addition to solve the system of equations.

次の各問について、足し算を使ってこの連立方程式を解け。

6. $\begin{array}{l} 3 x + 2 y = −7 \\ 2 x + 4 y = 6 \end{array}$

7. $\begin{array}{r} 3 x + 4 y = 2 \\ 9 x + 12 y = 3 \end{array}$

8. $\begin{array}{l} 8 x + 4 y = 2 \\ 6 x - 5 y = 0.7 \end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

次の各問について、各問題を解く連立方程式を書け。そしてその連立方程式を解け。

9. A factory has a cost of production $C (x) = 150 x + 15, 000$ and a revenue function $R (x) = 200 x .$ What is the break-even point?

9. ある工場の製造の費用が $C (x) = 150 x + 15, 000$ 、収入関数が $R (x) = 200 x$ である。損益分岐点はいくらか。

10. A performer charges $C (x) = 50 x + 10, 000,$ where $x$ is the total number of attendees at a show. The venue charges $75 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

10. ある演者は $C (x) = 50 x + 10, 000$ を請求する。ここで $x$ は公演の入場者数の合計である。会場は切符1枚あたり75ドルを取る。何人が切符を買えば会場は損得なしになり、その時点で売れた切符の総額はいくらか。

Systems of Linear Equations: Three Variables連立一次方程式: 三変数

For the following exercises, solve the system of three equations using substitution or addition.

次の各問について、代入または足し算を使って三式の組を解け。

11. $\begin{array}{l} 0.5 x - 0.5 y = 10 \\ - 0.2 y + 0.2 x = 4 \\ 0.1 x + 0.1 z = 2 \end{array}$

12. $\begin{array}{r} 5 x + 3 y - z = 5 \\ 3 x - 2 y + 4 z = 13 \\ 4 x + 3 y + 5 z = 22 \end{array}$

13. $\begin{array}{r} x + y + z = 1 \\ 2 x + 2 y + 2 z = 1 \\ 3 x + 3 y = 2 \end{array}$

14. $\begin{array}{l} 2 x - 3 y + z = −1 \\ x + y + z = −4 \\ 4 x + 2 y - 3 z = 33 \end{array}$

15. $\begin{array}{l} 3 x + 2 y - z = −10 \\ x - y + 2 z = 7 \\ - x + 3 y + z = −2 \end{array}$

16. $\begin{array}{r} 3 x + 4 z = −11 \\ x - 2 y = 5 \\ 4 y - z = −10 \end{array}$

17. $\begin{array}{r} 2 x - 3 y + z = 0 \\ 2 x + 4 y - 3 z = 0 \\ 6 x - 2 y - z = 0 \end{array}$

18. $\begin{array}{r} 6 x - 4 y - 2 z = 2 \\ 3 x + 2 y - 5 z = 4 \\ 6 y - 7 z = 5 \end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

次の各問について、各問題を解く連立方程式を書け。そしてその連立方程式を解け。

19. Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

19. 三つの奇数の和が61である。小さいほうは大きいほうの3分の1で、真ん中の数は大きいほうより16小さい。この三つの数はいくらか。

20. A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

20. ある地元の劇場の公演が満員になった。500枚の券をすべて売り、売り上げの合計は8,070.00ドルだった。券の値段は学生15ドル、子ども12ドル、大人18ドルだった。この楽団が子ども券の3倍の大人券を売ったとすると、各種類は何枚ずつ売れたか。

Systems of Nonlinear Equations and Inequalities: Two Variables非線形連立方程式と連立不等式: 二変数

For the following exercises, solve the system of nonlinear equations.

次の各問について、この非線形連立方程式を解け。

21. $\begin{array}{l} y = x^{2} - 7 \\ y = 5 x - 13 \end{array}$

22. $\begin{array}{l} y = x^{2} - 4 \\ y = 5 x + 10 \end{array}$

23. $\begin{array}{l} x^{2} + y^{2} = 16 \\ y = x - 8 \end{array}$

24. $\begin{array}{l} x^{2} + y^{2} = 25 \\ y = x^{2} + 5 \end{array}$

25. $\begin{array}{r} x^{2} + y^{2} = 4 \\ y - x^{2} = 3 \end{array}$

For the following exercises, graph the inequality.

次の各問について、この不等式のグラフを描け。

26. $y > x^{2} - 1$

27. $\frac{1}{4} x^{2} + y^{2} < 4$

For the following exercises, graph the system of inequalities.

次の各問について、この連立不等式のグラフを描け。

28. $\begin{array}{l} x^{2} + y^{2} + 2 x < 3 \\ y > - x^{2} - 3 \end{array}$

29. $\begin{array}{l} x^{2} - 2 x + y^{2} - 4 x < 4 \\ y < - x + 4 \end{array}$

30. $\begin{array}{l} x^{2} + y^{2} < 1 \\ y^{2} < x \end{array}$

Partial Fractions部分分数

For the following exercises, decompose into partial fractions.

次の各問について、部分分数に分解せよ。

31. $\frac{- 2 x + 6}{x^{2} + 3 x + 2}$

32. $\frac{10 x + 2}{4 x^{2} + 4 x + 1}$

33. $\frac{7 x + 20}{x^{2} + 10 x + 25}$

34. $\frac{x - 18}{x^{2} - 12 x + 36}$

35. $\frac{- x^{2} + 36 x + 70}{x^{3} - 125}$

36. $\frac{- 5 x^{2} + 6 x - 2}{x^{3} + 27}$

37. $\frac{x^{3} - 4 x^{2} + 3 x + 11}{{(x^{2} - 2)}^{2}}$

38. $\frac{4 x^{4} - 2 x^{3} + 22 x^{2} - 6 x + 48}{x {(x^{2} + 4)}^{2}}$

Matrices and Matrix Operations行列と行列の演算

For the following exercises, perform the requested operations on the given matrices.

次の各問について、与えられた行列に求められた演算を行え。

A = [\begin{array}{r} 4 & - 2 \\ 1 & 3 \end{array}], B = [\begin{array}{r} 6 & 7 & - 3 \\ 11 & - 2 & 4 \end{array}], C = [\begin{array}{r} \begin{matrix} 6 & 7 \\ 11 & - 2 \end{matrix} \\ \begin{matrix} 14 & 0 \end{matrix} \end{array}], D = [\begin{array}{r} 1 & - 4 & 9 \\ 10 & 5 & - 7 \\ 2 & 8 & 5 \end{array}], E = [\begin{array}{r} 7 & - 14 & 3 \\ 2 & - 1 & 3 \\ 0 & 1 & 9 \end{array}]

39. $- 4 A$

40. $10 D - 6 E$

41. $B + C$

42. $A B$

43. $B A$

44. $B C$

45. $C B$

46. $D E$

47. $E D$

48. $E C$

49. $C E$

50. $A^{3}$

Solving Systems with Gaussian Eliminationガウスの消去法で連立方程式を解く

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

次の各問について、この拡大係数行列から連立一次方程式を書け。解がただ一つあるかどうかを示せ。

51. $[\begin{array}{r} 1 & 0 & −3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array} | \begin{array}{r} 7 \\ −5 \\ 0 \end{array}]$

52. $[\begin{array}{r} 1 & 0 & 5 \\ 0 & 1 & −2 \\ 0 & 0 & 0 \end{array} | \begin{array}{r} −9 \\ 4 \\ 3 \end{array}]$

For the following exercises, write the augmented matrix from the system of linear equations.

次の各問について、この連立一次方程式から拡大係数行列を書け。

53. $\begin{array}{r} - 2 x + 2 y + z = 7 \\ 2 x - 8 y + 5 z = 0 \\ 19 x - 10 y + 22 z = 3 \end{array}$

54. $\begin{array}{l} 4 x + 2 y - 3 z = 14 \\ - 12 x + 3 y + z = 100 \\ 9 x - 6 y + 2 z = 31 \end{array}$

55. $\begin{array}{r} x + 3 z = 12 \\ - x + 4 y = 0 \\ y + 2 z = - 7 \end{array}$

For the following exercises, solve the system of linear equations using Gaussian elimination.

次の各問について、ガウスの消去法を使ってこの連立一次方程式を解け。

56. $\begin{array}{r} 3 x - 4 y = - 7 \\ - 6 x + 8 y = 14 \end{array}$

57. $\begin{array}{r} 3 x - 4 y = 1 \\ - 6 x + 8 y = 6 \end{array}$

58. $\begin{array}{l} - 1.1 x - 2.3 y = 6.2 \\ - 5.2 x - 4.1 y = 4.3 \end{array}$

59. $\begin{array}{r} 2 x + 3 y + 2 z = 1 \\ - 4 x - 6 y - 4 z = - 2 \\ 10 x + 15 y + 10 z = 0 \end{array}$

60. $\begin{array}{r} - x + 2 y - 4 z = 8 \\ 3 y + 8 z = - 4 \\ - 7 x + y + 2 z = 1 \end{array}$

Solving Systems with Inverses逆行列で連立方程式を解く

For the following exercises, find the inverse of the matrix.

次の各問について、この行列の逆を求めよ。

61. $[\begin{array}{r} - 0.2 & 1.4 \\ 1.2 & - 0.4 \end{array}]$

62. $[\begin{array}{r} \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{4} & \frac{3}{4} \end{array}]$

63. $[\begin{matrix} 12 & 9 & - 6 \\ - 1 & 3 & 2 \\ - 4 & - 3 & 2 \end{matrix}]$

64. $[\begin{matrix} 2 & 1 & 3 \\ 1 & 2 & 3 \\ 3 & 2 & 1 \end{matrix}]$

For the following exercises, find the solutions by computing the inverse of the matrix.

次の各問について、行列の逆を計算して解を求めよ。

65. $\begin{array}{l} 0.3 x - 0.1 y = - 10 \\ - 0.1 x + 0.3 y = 14 \end{array}$

66. $\begin{array}{l} 0.4 x - 0.2 y = - 0.6 \\ - 0.1 x + 0.05 y = 0.3 \end{array}$

67. $\begin{array}{r} 4 x + 3 y - 3 z = - 4.3 \\ 5 x - 4 y - z = - 6.1 \\ x + z = - 0.7 \end{array}$

68. $\begin{array}{r} \begin{array}{l} - 2 x - 3 y + 2 z = 3 \end{array} \\ - x + 2 y + 4 z = - 5 \\ - 2 y + 5 z = - 3 \end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

次の各問について、各問題を解く連立方程式を書け。そしてその連立方程式を解け。

69. Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?

69. 学生たちが好きな果物を教室に持ってくるよう言われた。果物の90%はバナナ、りんご、みかんだった。みかんがバナナの半分の人気で、りんごがバナナより5%人気が高かったとすると、各果物の割合はそれぞれ何パーセントか。

70. A school club held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $2 and the chocolate chip cookies at $1. They raised $250 and sold 175 items. How many brownies and how many cookies were sold?

70. ある学校のクラブが資金集めの菓子の販売を開き、ブラウニーとチョコチップクッキーを売った。ブラウニーを2ドル、チョコチップクッキーを1ドルの値段にした。250ドルを集め、175個を売った。ブラウニーとクッキーは何個ずつ売れたか。

Solving Systems with Cramer's Ruleクラメルの公式で連立方程式を解く

For the following exercises, find the determinant.

次の各問について、行列式を求めよ。

71. $| \begin{matrix} 100 & 0 \\ 0 & 0 \end{matrix} |$

72. $| \begin{matrix} 0.2 & - 0.6 \\ 0.7 & - 1.1 \end{matrix} |$

73. $| \begin{matrix} - 1 & 4 & 3 \\ 0 & 2 & 3 \\ 0 & 0 & - 3 \end{matrix} |$

74. $| \begin{matrix} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{matrix} |$

For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

次の各問について、クラメルの公式を使ってこの連立一次方程式を解け。

75. $\begin{array}{r} 4 x - 2 y = 23 \\ - 5 x - 10 y = - 35 \end{array}$

76. $\begin{array}{l} 0.2 x - 0.1 y = 0 \\ - 0.3 x + 0.3 y = 2.5 \end{array}$

77. $\begin{array}{r} - 0.5 x + 0.1 y = 0.3 \\ - 0.25 x + 0.05 y = 0.15 \end{array}$

78. $\begin{array}{l} x + 6 y + 3 z = 4 \\ 2 x + y + 2 z = 3 \\ 3 x - 2 y + z = 0 \end{array}$

79. $\begin{array}{r} 4 x - 3 y + 5 z = - \frac{5}{2} \\ 7 x - 9 y - 3 z = \frac{3}{2} \\ x - 5 y - 5 z = \frac{5}{2} \end{array}$

80. $\begin{array}{r} \frac{3}{10} x - \frac{1}{5} y - \frac{3}{10} z = - \frac{1}{50} \\ \frac{1}{10} x - \frac{1}{10} y - \frac{1}{2} z = - \frac{9}{50} \\ \frac{2}{5} x - \frac{1}{2} y - \frac{3}{5} z = - \frac{1}{5} \end{array}$