Solutions 32: Solving Systems with Gaussian Elimination

Solutions to Try Its

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

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

3.
$\left(2,1\right)$

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

5.
$\left(1,1,1\right)$

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 ${R}_{2}={R}_{2}-9{R}_{1}$ . (2) ${R}_{2}={R}_{1}-9{R}_{2}$ . 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. $\left[\begin{array}{rrrr}\qquad 0& \qquad & \qquad 16& \qquad \\ \qquad 9& \qquad & \qquad -1& \qquad \end{array}|\begin{array}{rr}\qquad & \qquad 4\\ \qquad & \qquad 2\end{array}\right]$ 9. $\left[\begin{array}{rrrrrr}\qquad 1& \qquad & \qquad 5& \qquad & \qquad 8& \qquad \\ \qquad 12& \qquad & \qquad 3& \qquad & \qquad 0& \qquad \\ \qquad 3& \qquad & \qquad 4& \qquad & \qquad 9& \qquad \end{array}|\begin{array}{rr}\qquad & \qquad 16\\ \qquad & \qquad 4\\ \qquad & \qquad -7\end{array}\right]$ 11. $\begin{array}{l}-2x+5y=5\\ 6x - 18y=26\end{array}$ 13. $\begin{array}{l}3x+2y=13\\ -x - 9y+4z=53\\ 8x+5y+7z=80\end{array}$ 15. $\begin{array}{l}4x+5y - 2z=12\qquad \\ \text{ }y+58z=2\qquad \\ 8x+7y - 3z=-5\qquad \end{array}$ 17. No solutions 19. $\left(-1,-2\right)$ 21. $\left(6,7\right)$ 23. $\left(3,2\right)$ 25. $\left(\frac{1}{5},\frac{1}{2}\right)$ 27. $\left(x,\frac{4}{15}\left(5x+1\right)\right)$ 29. $\left(3,4\right)$ 31. $\left(\frac{196}{39},-\frac{5}{13}\right)$ 33. $\left(31,-42,87\right)$ 35. $\left(\frac{21}{40},\frac{1}{20},\frac{9}{8}\right)$ 37. $\left(\frac{18}{13},\frac{15}{13},-\frac{15}{13}\right)$ 39. $\left(x,y,\frac{1}{2}\left(1 - 2x - 3y\right)\right)$ 41. $\left(x,-\frac{x}{2},-1\right)$ 43. $\left(125,-25,0\right)$ 45. $\left(8,1,-2\right)$ 47. $\left(1,2,3\right)$ 49. $\left(x,\frac{31}{28}-\frac{3x}{4},\frac{1}{28}\left(-7x - 3\right)\right)$ 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