Problem Set 32: Solving Systems with Gaussian Elimination

1. Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

2. Can any matrix be written as a system of linear equations? Explain why or why not. Explain how to write that system of equations.

3. Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix [latex]\left[\begin{array}{rr}\hfill 9& \hfill 3\\ \hfill 1& \hfill -2\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 0\\ \hfill 6\end{array}\right][/latex].

4. Can a matrix whose entry is 0 on the diagonal be solved? Explain why or why not. What would you do to remedy the situation?

5. Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.

For the following exercises, write the augmented matrix for the linear system.

6. [latex]\begin{array}{l}8x - 37y=8\\ 2x+12y=3\end{array}[/latex]

7. [latex]\begin{array}{l}\text{ }16y=4\hfill \\ 9x-y=2\hfill \end{array}[/latex]

8. [latex]\begin{array}{l}\text{ }3x+2y+10z=3\hfill \\ -6x+2y+5z=13\hfill \\ \text{ }4x+z=18\hfill \end{array}[/latex]

9. [latex]\begin{array}{l}\hfill \\ \text{ }x+5y+8z=19\hfill \\ \text{ }12x+3y=4\hfill \\ 3x+4y+9z=-7\hfill \end{array}[/latex]

10. [latex]\begin{array}{l}6x+12y+16z=4\hfill \\ \text{ }19x - 5y+3z=-9\hfill \\ \text{ }x+2y=-8\hfill \end{array}[/latex]

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

11. [latex]\left[\begin{array}{rr}\hfill -2& \hfill 5\\ \hfill 6& \hfill -18\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 5\\ \hfill 26\end{array}\right][/latex]

12. [latex]\left[\begin{array}{rr}\hfill 3& \hfill 4\\ \hfill 10& \hfill 17\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 10\\ \hfill 439\end{array}\right][/latex]

13. [latex]\left[\begin{array}{rrr}\hfill 3& \hfill 2& \hfill 0\\ \hfill -1& \hfill -9& \hfill 4\\ \hfill 8& \hfill 5& \hfill 7\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 3\\ \hfill -1\\ \hfill 8\end{array}\right][/latex]

14. [latex]\left[\begin{array}{rrr}\hfill 8& \hfill 29& \hfill 1\\ \hfill -1& \hfill 7& \hfill 5\\ \hfill 0& \hfill 0& \hfill 3\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 43\\ \hfill 38\\ \hfill 10\end{array}\right][/latex]

15. [latex]\left[\begin{array}{rrr}\hfill 4& \hfill 5& \hfill -2\\ \hfill 0& \hfill 1& \hfill 58\\ \hfill 8& \hfill 7& \hfill -3\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 12\\ \hfill 2\\ \hfill -5\end{array}\right][/latex]

For the following exercises, solve the system by Gaussian elimination.

16. [latex]\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 0& \hfill 0\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 3\\ \hfill 0\end{array}\right][/latex]

17. [latex]\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 1& \hfill 0\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 1\\ \hfill 2\end{array}\right][/latex]

18. [latex]\left[\begin{array}{rr}\hfill 1& \hfill 2\\ \hfill 4& \hfill 5\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 3\\ \hfill 6\end{array}\right][/latex]

19. [latex]\left[\begin{array}{rr}\hfill -1& \hfill 2\\ \hfill 4& \hfill -5\end{array}\text{ }|\text{ }\begin{array}{r}\hfill -3\\ \hfill 6\end{array}\right][/latex]

20. [latex]\left[\begin{array}{rr}\hfill -2& \hfill 0\\ \hfill 0& \hfill 2\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 1\\ \hfill -1\end{array}\right][/latex]

21. [latex]\begin{array}{l}\text{ }2x - 3y=-9\hfill \\ 5x+4y=58\hfill \end{array}[/latex]

22. [latex]\begin{array}{l}6x+2y=-4\\ 3x+4y=-17\end{array}[/latex]

23. [latex]\begin{array}{l}2x+3y=12\hfill \\ \text{ }4x+y=14\hfill \end{array}[/latex]

24. [latex]\begin{array}{l}-4x - 3y=-2\hfill \\ \text{ }3x - 5y=-13\hfill \end{array}[/latex]

25. [latex]\begin{array}{l}-5x+8y=3\hfill \\ 10x+6y=5\hfill \end{array}[/latex]

26. [latex]\begin{array}{l}\text{ }3x+4y=12\hfill \\ -6x - 8y=-24\hfill \end{array}[/latex]

27. [latex]\begin{array}{l}-60x+45y=12\hfill \\ \text{ }20x - 15y=-4\hfill \end{array}[/latex]

28. [latex]\begin{array}{l}11x+10y=43\\ 15x+20y=65\end{array}[/latex]

29. [latex]\begin{array}{l}\text{ }2x-y=2\hfill \\ 3x+2y=17\hfill \end{array}[/latex]

30. [latex]\begin{array}{l}\begin{array}{l}\\ -1.06x - 2.25y=5.51\end{array}\hfill \\ -5.03x - 1.08y=5.40\hfill \end{array}[/latex]

31. [latex]\begin{array}{l}\frac{3}{4}x-\frac{3}{5}y=4\\ \frac{1}{4}x+\frac{2}{3}y=1\end{array}[/latex]

32. [latex]\begin{array}{l}\frac{1}{4}x-\frac{2}{3}y=-1\\ \frac{1}{2}x+\frac{1}{3}y=3\end{array}[/latex]

33. [latex]\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 1\\ \hfill 0& \hfill 0& \hfill 1\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 31\\ \hfill 45\\ \hfill 87\end{array}\right][/latex]

34. [latex]\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 1\\ \hfill 1& \hfill 1& \hfill 0\\ \hfill 0& \hfill 1& \hfill 1\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 50\\ \hfill 20\\ \hfill -90\end{array}\right][/latex]

35. [latex]\left[\begin{array}{rrr}\hfill 1& \hfill 2& \hfill 3\\ \hfill 0& \hfill 5& \hfill 6\\ \hfill 0& \hfill 0& \hfill 8\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 4\\ \hfill 7\\ \hfill 9\end{array}\right][/latex]

36. [latex]\left[\begin{array}{rrr}\hfill -0.1& \hfill 0.3& \hfill -0.1\\ \hfill -0.4& \hfill 0.2& \hfill 0.1\\ \hfill 0.6& \hfill 0.1& \hfill 0.7\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 0.2\\ \hfill 0.8\\ \hfill -0.8\end{array}\right][/latex]

37. [latex]\begin{array}{l}\text{ }-2x+3y - 2z=3\hfill \\ \text{ }4x+2y-z=9\hfill \\ \text{ }4x - 8y+2z=-6\hfill \end{array}[/latex]

38. [latex]\begin{array}{l}\text{ }x+y - 4z=-4\hfill \\ \text{ }5x - 3y - 2z=0\hfill \\ \text{ }2x+6y+7z=30\hfill \end{array}[/latex]

39. [latex]\begin{array}{l}\text{ }2x+3y+2z=1\hfill \\ \text{ }-4x - 6y - 4z=-2\hfill \\ \text{ }10x+15y+10z=5\hfill \end{array}[/latex]

40. [latex]\begin{array}{l}\text{ }x+2y-z=1\hfill \\ -x - 2y+2z=-2\hfill \\ 3x+6y - 3z=5\hfill \end{array}[/latex]

41. [latex]\begin{array}{l}\text{ }x+2y-z=1\hfill \\ -x - 2y+2z=-2\hfill \\ \text{ }3x+6y - 3z=3\hfill \end{array}[/latex]

42. [latex]\begin{array}{l}\text{ }\text{ }x+y=2\hfill \\ \text{ }x+z=1\hfill \\ -y-z=-3\hfill \end{array}[/latex]

43. [latex]\begin{array}{l}x+y+z=100\hfill \\ \text{ }x+2z=125\hfill \\ -y+2z=25\hfill \end{array}[/latex]

44. [latex]\begin{array}{l}\frac{1}{4}x-\frac{2}{3}z=-\frac{1}{2}\\ \frac{1}{5}x+\frac{1}{3}y=\frac{4}{7}\\ \frac{1}{5}y-\frac{1}{3}z=\frac{2}{9}\end{array}[/latex]

45. [latex]\begin{array}{l}-\frac{1}{2}x+\frac{1}{2}y+\frac{1}{7}z=-\frac{53}{14}\hfill \\ \text{ }\frac{1}{2}x-\frac{1}{2}y+\frac{1}{4}z=3\hfill \\ \text{ }\frac{1}{4}x+\frac{1}{5}y+\frac{1}{3}z=\frac{23}{15}\hfill \end{array}[/latex]

46. [latex]\begin{array}{l}-\frac{1}{2}x-\frac{1}{3}y+\frac{1}{4}z=-\frac{29}{6}\hfill \\ \text{ }\frac{1}{5}x+\frac{1}{6}y-\frac{1}{7}z=\frac{431}{210}\hfill \\ -\frac{1}{8}x+\frac{1}{9}y+\frac{1}{10}z=-\frac{49}{45}\hfill \end{array}[/latex]

For the following exercises, use Gaussian elimination to solve the system.

47. [latex]\begin{array}{l}\frac{x - 1}{7}+\frac{y - 2}{8}+\frac{z - 3}{4}=0\hfill \\ \text{ }x+y+z=6\hfill \\ \text{ }\frac{x+2}{3}+2y+\frac{z - 3}{3}=5\hfill \end{array}[/latex]

48. [latex]\begin{array}{l}\frac{x - 1}{4}-\frac{y+1}{4}+3z=-1\hfill \\ \text{ }\frac{x+5}{2}+\frac{y+7}{4}-z=4\hfill \\ \text{ }x+y-\frac{z - 2}{2}=1\hfill \end{array}[/latex]

49. [latex]\begin{array}{l}\text{ }\frac{x - 3}{4}-\frac{y - 1}{3}+2z=-1\hfill \\ \frac{x+5}{2}+\frac{y+5}{2}+\frac{z+5}{2}=8\hfill \\ \text{ }x+y+z=1\hfill \end{array}[/latex]

50. [latex]\begin{array}{l}\frac{x - 3}{10}+\frac{y+3}{2}-2z=3\hfill \\ \text{ }\frac{x+5}{4}-\frac{y - 1}{8}+z=\frac{3}{2}\hfill \\ \frac{x - 1}{4}+\frac{y+4}{2}+3z=\frac{3}{2}\hfill \end{array}[/latex]

51. [latex]\begin{array}{l}\text{ }\frac{x - 3}{4}-\frac{y - 1}{3}+2z=-1\hfill \\ \frac{x+5}{2}+\frac{y+5}{2}+\frac{z+5}{2}=7\hfill \\ \text{ }x+y+z=1\hfill \end{array}[/latex]

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.

52. Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

53. At a competing cupcake store, $4,520 worth of cupcakes are sold daily. The chocolate cupcakes cost $2.25 and the red velvet cupcakes cost $1.75. If the total number of cupcakes sold per day is 2,200, how many of each flavor are sold each day?

54. You invested $10,000 into two accounts: one that has simple 3% interest, the other with 2.5% interest. If your total interest payment after one year was $283.50, how much was in each account after the year passed?

55. You invested $2,300 into account 1, and $2,700 into account 2. If the total amount of interest after one year is $254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

56. Bikes’R’Us manufactures bikes, which sell for $250. It costs the manufacturer $180 per bike, plus a startup fee of $3,500. After how many bikes sold will the manufacturer break even?

57. A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for $86 each, with a delivery fee of $9,200, regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?

58. The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising 83% of the flavors sold at an ice cream shop. If vanilla sells 1% more than twice strawberry, and chocolate sells 11% more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

59. At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

60. A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 g, pistachios weigh 4 g, and almonds weigh 5 g. If the bag weighs 3.7 kg, find out how many of each type of nut is in the bag.

61. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.