Solutions 32: Solving Systems with Gaussian Elimination

Solutions to Try Its

1. 
[4332114]\left[\begin{array}{cc}4& -3\\ 3& 2\end{array}|\begin{array}{c}11\\ 4\end{array}\right]


2. 
xy+z=52xy+3z=1y+z=9\begin{array}{c}x-y+z=5\\ 2x-y+3z=1\\ y+z=-9\end{array}


3. 
(2,1)\left(2,1\right)


4. 
[15252 01500117292]\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]


5. 
(1,1,1)\left(1,1,1\right)


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
R2=R29R1{R}_{2}={R}_{2}-9{R}_{1}
. (2)
R2=R19R2{R}_{2}={R}_{1}-9{R}_{2}
. 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. 
[0169142]\left[\begin{array}{rrrr}\qquad 0& \qquad & \qquad 16& \qquad \\ \qquad 9& \qquad & \qquad -1& \qquad \end{array}|\begin{array}{rr}\qquad & \qquad 4\\ \qquad & \qquad 2\end{array}\right]


9. 
[15812303491647]\left[\begin{array}{rrrrrr}\qquad 1& \qquad & \qquad 5& \qquad & \qquad 8& \qquad \\ \qquad 12& \qquad & \qquad 3& \qquad & \qquad 0& \qquad \\ \qquad 3& \qquad & \qquad 4& \qquad & \qquad 9& \qquad \end{array}|\begin{array}{rr}\qquad & \qquad 16\\ \qquad & \qquad 4\\ \qquad & \qquad -7\end{array}\right]


11. 
2x+5y=56x18y=26\begin{array}{l}-2x+5y=5\\ 6x - 18y=26\end{array}


13. 
3x+2y=13x9y+4z=538x+5y+7z=80\begin{array}{l}3x+2y=13\\ -x - 9y+4z=53\\ 8x+5y+7z=80\end{array}


15. 
4x+5y2z=12 y+58z=28x+7y3z=5\begin{array}{l}4x+5y - 2z=12\qquad \\ \text{ }y+58z=2\qquad \\ 8x+7y - 3z=-5\qquad \end{array}


17. No solutions

19. 
(1,2)\left(-1,-2\right)


21. 
(6,7)\left(6,7\right)


23. 
(3,2)\left(3,2\right)


25. 
(15,12)\left(\frac{1}{5},\frac{1}{2}\right)


27. 
(x,415(5x+1))\left(x,\frac{4}{15}\left(5x+1\right)\right)


29. 
(3,4)\left(3,4\right)


31. 
(19639,513)\left(\frac{196}{39},-\frac{5}{13}\right)


33. 
(31,42,87)\left(31,-42,87\right)


35. 
(2140,120,98)\left(\frac{21}{40},\frac{1}{20},\frac{9}{8}\right)


37. 
(1813,1513,1513)\left(\frac{18}{13},\frac{15}{13},-\frac{15}{13}\right)


39. 
(x,y,12(12x3y))\left(x,y,\frac{1}{2}\left(1 - 2x - 3y\right)\right)


41. 
(x,x2,1)\left(x,-\frac{x}{2},-1\right)


43. 
(125,25,0)\left(125,-25,0\right)


45. 
(8,1,2)\left(8,1,-2\right)


47. 
(1,2,3)\left(1,2,3\right)


49. 
(x,31283x4,128(7x3))\left(x,\frac{31}{28}-\frac{3x}{4},\frac{1}{28}\left(-7x - 3\right)\right)


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

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