Solutions for Solving Systems with Gaussian Elimination

Solutions to Try Its

1. [latex]\left[\begin{array}{cc}4& -3\\ 3& 2\end{array}|\begin{array}{c}11\\ 4\end{array}\right][/latex]

2. [latex]\begin{array}{c}x-y+z=5\\ 2x-y+3z=1\\ y+z=-9\end{array}[/latex]

3. [latex]\left(2,1\right)[/latex]

4. [latex]\left[\begin{array}{ccc}1& -\frac{5}{2}& \frac{5}{2}\\ \text{ }0& 1& 5\\ 0& 0& 1\end{array}|\begin{array}{c}\frac{17}{2}\\ 9\\ 2\end{array}\right][/latex]

5. [latex]\left(1,1,1\right)[/latex]

6. $150,000 at 7%, $750,000 at 8%, $600,000 at 10%

Solutions to Odd-Numbered Exercises

1. Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.

3. No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then [latex]{R}_{2}={R}_{2}-9{R}_{1}[/latex]. (2) [latex]{R}_{2}={R}_{1}-9{R}_{2}[/latex]. Then divide row 1 by 9.

5. No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions.

7. [latex]\left[\begin{array}{rrrr}\hfill 0& \hfill & \hfill 16& \hfill \\ \hfill 9& \hfill & \hfill -1& \hfill \end{array}|\begin{array}{rr}\hfill & \hfill 4\\ \hfill & \hfill 2\end{array}\right][/latex]

9. [latex]\left[\begin{array}{rrrrrr}\hfill 1& \hfill & \hfill 5& \hfill & \hfill 8& \hfill \\ \hfill 12& \hfill & \hfill 3& \hfill & \hfill 0& \hfill \\ \hfill 3& \hfill & \hfill 4& \hfill & \hfill 9& \hfill \end{array}|\begin{array}{rr}\hfill & \hfill 16\\ \hfill & \hfill 4\\ \hfill & \hfill -7\end{array}\right][/latex]

11. [latex]\begin{array}{l}-2x+5y=5\\ 6x - 18y=26\end{array}[/latex]

13. [latex]\begin{array}{l}3x+2y=13\\ -x - 9y+4z=53\\ 8x+5y+7z=80\end{array}[/latex]

15. [latex]\begin{array}{l}4x+5y - 2z=12\hfill \\ \text{ }y+58z=2\hfill \\ 8x+7y - 3z=-5\hfill \end{array}[/latex]

17. No solutions

19. [latex]\left(-1,-2\right)[/latex]

21. [latex]\left(6,7\right)[/latex]

23. [latex]\left(3,2\right)[/latex]

25. [latex]\left(\frac{1}{5},\frac{1}{2}\right)[/latex]

27. [latex]\left(x,\frac{4}{15}\left(5x+1\right)\right)[/latex]

29. [latex]\left(3,4\right)[/latex]

31. [latex]\left(\frac{196}{39},-\frac{5}{13}\right)[/latex]

33. [latex]\left(31,-42,87\right)[/latex]

35. [latex]\left(\frac{21}{40},\frac{1}{20},\frac{9}{8}\right)[/latex]

37. [latex]\left(\frac{18}{13},\frac{15}{13},-\frac{15}{13}\right)[/latex]

39. [latex]\left(x,y,\frac{1}{2}\left(1 - 2x - 3y\right)\right)[/latex]

41. [latex]\left(x,-\frac{x}{2},-1\right)[/latex]

43. [latex]\left(125,-25,0\right)[/latex]

45. [latex]\left(8,1,-2\right)[/latex]

47. [latex]\left(1,2,3\right)[/latex]

49. [latex]\left(x,\frac{31}{28}-\frac{3x}{4},\frac{1}{28}\left(-7x - 3\right)\right)[/latex]

51. No solutions exist.

53. 860 red velvet, 1,340 chocolate

55. 4% for account 1, 6% for account 2

57. $126

59. Banana was 3%, pumpkin was 7%, and rocky road was 2%

61. 100 almonds, 200 cashews, 600 pistachios