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

Practice Test実力テスト

Is the following ordered pair a solution to the system of equations?

次の順序対はこの連立方程式の解か。

1. $\begin{array}{l} \begin{array}{l} - 5 x - y = 12 \\ x + 4 y = 9 \end{array} \end{array}$ with $(- 3, 3)$

1. $(- 3, 3)$ のときの $\begin{array}{l} \begin{array}{l} - 5 x - y = 12 \\ x + 4 y = 9 \end{array} \end{array}$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

次の各問について、代入または消去を使って、一次および非線形連立方程式を解け。解が無いならそう示せ。

2. $\begin{array}{r} \frac{1}{2} x - \frac{1}{3} y = 4 \\ \frac{3}{2} x - y = 0 \end{array}$

3. $\begin{array}{l} - \frac{1}{2} x - 4 y = 4 \\ 2 x + 16 y = 2 \end{array}$

4. $\begin{array}{r} 5 x - y = 1 \\ - 10 x + 2 y = - 2 \end{array}$

5. $\begin{array}{l} 4 x - 6 y - 2 z = \frac{1}{10} \\ x - 7 y + 5 z = - \frac{1}{4} \\ 3 x + 6 y - 9 z = \frac{6}{5} \end{array}$

6. $\begin{array}{r} x + z = 20 \\ x + y + z = 20 \\ x + 2 y + z = 10 \end{array}$

7. $\begin{array}{r} 5 x - 4 y - 3 z = 0 \\ 2 x + y + 2 z = 0 \\ x - 6 y - 7 z = 0 \end{array}$

8. $\begin{array}{l} y = x^{2} + 2 x - 3 \\ y = x - 1 \end{array}$

9. $\begin{array}{l} y^{2} + x^{2} = 25 \\ y^{2} - 2 x^{2} = 1 \end{array}$

For the following exercises, graph the following inequalities.

次の各問について、次の不等式のグラフを描け。

10. $y < x^{2} + 9$

11. $\begin{array}{l} x^{2} + y^{2} > 4 \\ y < x^{2} + 1 \end{array}$

For the following exercises, write the partial fraction decomposition.

次の各問について、部分分数分解を書け。

12. $\frac{- 8 x - 30}{x^{2} + 10 x + 25}$

13. $\frac{13 x + 2}{{(3 x + 1)}^{2}}$

14. $\frac{x^{4} - x^{3} + 2 x - 1}{x {(x^{2} + 1)}^{2}}$

For the following exercises, perform the given matrix operations.

次の各問について、与えられた行列の演算を行え。

15. $5 [\begin{matrix} 4 & 9 \\ - 2 & 3 \end{matrix}] + \frac{1}{2} [\begin{matrix} - 6 & 12 \\ 4 & - 8 \end{matrix}]$

16. $[\begin{array}{r} 1 & 4 & - 7 \\ - 2 & 9 & 5 \\ 12 & 0 & - 4 \end{array}] [\begin{matrix} 3 & - 4 \\ 1 & 3 \\ 5 & 10 \end{matrix}]$

17. ${[\begin{array}{r} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{4} & \frac{1}{5} \end{array}]}^{- 1}$

18. $\det | \begin{matrix} 0 & 0 \\ 400 & 4, 000 \end{matrix} |$

19. $\det | \begin{array}{r} \frac{1}{2} & - \frac{1}{2} & 0 \\ - \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0 \end{array} |$

20. If $\det (A) = −6,$ what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?

20. $\det (A) = −6$ のとき、1行目と3行目を入れ替え、2行目に12を掛け、逆を取ったなら、行列式はいくらになるか。

21. Rewrite the system of linear equations as an augmented matrix.

21. この連立一次方程式を拡大係数行列として書き直せ。

\begin{array}{l} 14 x - 2 y + 13 z = 140 \\ - 2 x + 3 y - 6 z = - 1 \\ x - 5 y + 12 z = 11 \end{array}

22. Rewrite the augmented matrix as a system of linear equations.

22. この拡大係数行列を連立一次方程式として書き直せ。

[\begin{array}{r} 1 & 0 & 3 \\ - 2 & 4 & 9 \\ - 6 & 1 & 2 \end{array} | \begin{array}{r} 12 \\ - 5 \\ 8 \end{array}]

For the following exercises, use Gaussian elimination to solve the systems of equations.

次の各問について、ガウスの消去法を使ってこの連立方程式を解け。

23. $\begin{array}{r} x - 6 y = 4 \\ 2 x - 12 y = 0 \end{array}$

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

For the following exercises, use the inverse of a matrix to solve the systems of equations.

次の各問について、行列の逆を使ってこの連立方程式を解け。

25. $\begin{array}{r} 4 x - 5 y = - 50 \\ - x + 2 y = 80 \end{array}$

26. $\begin{array}{r} \frac{1}{100} x - \frac{3}{100} y + \frac{1}{20} z = - 49 \\ \frac{3}{100} x - \frac{7}{100} y - \frac{1}{100} z = 13 \\ \frac{9}{100} x - \frac{9}{100} y - \frac{9}{100} z = 99 \end{array}$

For the following exercises, use Cramer’s Rule to solve the 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方程式を解け。

27. $\begin{array}{l} 200 x - 300 y = 2 \\ 400 x + 715 y = 4 \end{array}$

28. $\begin{array}{l} 0.1 x + 0.1 y - 0.1 z = - 1.2 \\ 0.1 x - 0.2 y + 0.4 z = - 1.2 \\ 0.5 x - 0.3 y + 0.8 z = - 5.9 \end{array}$

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

次の各問について、連立一次方程式を使って解け。

29. A factory producing cell phones has the following cost and revenue functions: $C (x) = x^{2} + 75 x + 2, 688$ and $R (x) = x^{2} + 160 x .$ What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

29. 携帯電話を作るある工場の費用関数と収入関数は次である。 $C (x) = x^{2} + 75 x + 2, 688$ と $R (x) = x^{2} + 160 x$ 。利益が出るために毎日作るべき携帯電話の台数の範囲はいくらか。利益が出る整数に丸めよ。

30. A small fair charges $1.50 for students, $1 for children, and $2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of $1,050. How many of each type of ticket was sold?

30. ある小さな祭りは、学生1.50ドル、子ども1ドル、大人2ドルを取る。ある日、子どもは大人の3倍来場した。券は合わせて800枚売れ、収入の合計は1,050ドルだった。各種類の券は何枚ずつ売れたか。