1. Can a linear system of three equations have exactly two solutions? Explain why or why not<\/p>\n
2.\u00a0If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.<\/p>\n
3. If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.<\/p>\n
4. Using the method of addition, is there only one way to solve the system?<\/p>\n
5. Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.<\/p>\n
For the following exercises, determine whether the ordered triple given is the solution to the system of equations.<\/p>\n
6. [latex]\\begin{align}2x - 6y+6z&=-12\\\\x+4y+5z&=-1 \\\\ -x+2y+3z&=-1 \\end{align}[\/latex] and [latex]\\left(0,1,-1\\right)[\/latex]<\/p>\n
7. [latex]\\begin{align}6x-y+3z&=6 \\\\ 3x+5y+2z&=0 \\\\ x+y&=0 \\end{align}[\/latex] and [latex]\\left(3,-3,-5\\right)[\/latex]<\/p>\n
8.\u00a0[latex]\\begin{align}6x - 7y+z&=2 \\\\ -x-y+3z&=4 \\\\ 2x+y-z&=1 \\end{align}[\/latex] and [latex]\\left(4,2,-6\\right)[\/latex]<\/p>\n
9. [latex]\\begin{align}x-y&=0 \\\\ x-z&=5 \\\\ x-y+z&=-1 \\end{align}[\/latex] and [latex]\\left(4,4,-1\\right)[\/latex]<\/p>\n
10. [latex]\\begin{align} -x-y+2z&=3 \\\\ 5x+8y - 3z&=4 \\\\ -x+3y - 5z&=-5 \\end{align}[\/latex] and [latex]\\left(4,1,-7\\right)[\/latex]<\/p>\n
For the following exercises, solve each system by substitution.<\/p>\n
11. [latex]\\begin{align}3x - 4y+2z&=-15 \\\\ 2x+4y+z&=16 \\\\ 2x+3y+5z&=20 \\end{align}[\/latex]<\/p>\n
12.\u00a0[latex]\\begin{align}5x - 2y+3z&=20 \\\\ 2x - 4y - 3z&=-9 \\\\ x+6y - 8z&=21 \\end{align}[\/latex]<\/p>\n
13. [latex]\\begin{align}5x+2y+4z&=9 \\\\ -3x+2y+z&=10 \\\\ 4x - 3y+5z&=-3 \\end{align}[\/latex]<\/p>\n
14.\u00a0[latex]\\begin{align}4x - 3y+5z&=31 \\\\ -x+2y+4z&=20 \\\\ x+5y - 2z&=-29 \\end{align}[\/latex]<\/p>\n
15. [latex]\\begin{align}5x - 2y+3z&=4 \\\\ -4x+6y - 7z&=-1 \\\\ 3x+2y-z&=4\\end{align}[\/latex]<\/p>\n
16.\u00a0[latex]\\begin{align} 4x+6y+9z&=0 \\\\ -5x+2y - 6z&=3 \\\\ 7x - 4y+3z&=-3 \\end{align}[\/latex]<\/p>\n
For the following exercises, solve each system by Gaussian elimination.<\/p>\n
17. [latex]\\begin{align}2x-y+3z&=17 \\\\ -5x+4y - 2z&=-46 \\\\ 2y+5z&=-7 \\end{align}[\/latex]<\/p>\n
18.\u00a0[latex]\\begin{align}5x - 6y+3z&=50 \\\\ -x+4y&=10 \\\\ 2x-z&=10 \\end{align}[\/latex]<\/p>\n
19. [latex]\\begin{align}2x+3y - 6z&=1 \\\\ -4x - 6y+12z&=-2 \\\\ x+2y+5z&=10 \\end{align}[\/latex]<\/p>\n
20.\u00a0[latex]\\begin{align}4x+6y - 2z&=8 \\\\ 6x+9y - 3z&=12 \\\\ -2x - 3y+z&=-4 \\end{align}[\/latex]<\/p>\n
21. [latex]\\begin{align}2x+3y - 4z&=5 \\\\ -3x+2y+z&=11 \\\\ -x+5y+3z&=4 \\end{align}[\/latex]<\/p>\n
22.\u00a0[latex]\\begin{align}10x+2y - 14z&=8 \\\\ -x-2y - 4z&=-1 \\\\ -12x - 6y+6z&=-12 \\end{align}[\/latex]<\/p>\n
23. [latex]\\begin{align}x+y+z&=14 \\\\ 2y+3z&=-14 \\\\ -16y - 24z&=-112 \\end{align}[\/latex]<\/p>\n
24.\u00a0[latex]\\begin{align}5x - 3y+4z&=-1 \\\\ -4x+2y - 3z&=0 \\\\ -x+5y+7z&=-11 \\end{align}[\/latex]<\/p>\n
25. [latex]\\begin{align}x+y+z&=0 \\\\ 2x-y+3z&=0 \\\\ x-z&=0 \\end{align}[\/latex]<\/p>\n
26.\u00a0[latex]\\begin{align}3x+2y - 5z&=6\\\\ 5x - 4y+3z&=-12\\\\ 4x+5y - 2z&=15\\end{align}[\/latex]<\/p>\n
27. [latex]\\begin{align}x+y+z&=0\\\\ 2x-y+3z&=0 \\\\ x-z&=1 \\end{align}[\/latex]<\/p>\n
28.\u00a0[latex]\\begin{align} 3x-\\frac{1}{2}y-z&=-\\frac{1}{2} \\\\ 4x+z&=3 \\\\ -x+\\frac{3}{2}y&=\\frac{5}{2} \\end{align}[\/latex]<\/p>\n
29. [latex]\\begin{align}6x - 5y+6z&=38 \\\\ \\frac{1}{5}x-\\frac{1}{2}y+\\frac{3}{5}z&=1 \\\\ -4x-\\frac{3}{2}y-z&=-74 \\end{align}[\/latex]<\/p>\n
30.\u00a0[latex]\\begin{align}\\frac{1}{2}x-\\frac{1}{5}y+\\frac{2}{5}z&=-\\frac{13}{10} \\\\ \\frac{1}{4}x-\\frac{2}{5}y-\\frac{1}{5}z&=-\\frac{7}{20} \\\\ -\\frac{1}{2}x-\\frac{3}{4}y-\\frac{1}{2}z&=-\\frac{5}{4} \\end{align}[\/latex]<\/p>\n
31. [latex]\\begin{align} -\\frac{1}{3}x-\\frac{1}{2}y-\\frac{1}{4}z&=\\frac{3}{4} \\\\ -\\frac{1}{2}x-\\frac{1}{4}y-\\frac{1}{2}z&=2 \\\\ -\\frac{1}{4}x-\\frac{3}{4}y-\\frac{1}{2}z&=-\\frac{1}{2} \\end{align}[\/latex]<\/p>\n
32.\u00a0[latex]\\begin{align}\\frac{1}{2}x-\\frac{1}{4}y+\\frac{3}{4}z&=0\\\\ \\frac{1}{4}x-\\frac{1}{10}y+\\frac{2}{5}z&=-2\\\\ \\frac{1}{8}x+\\frac{1}{5}y-\\frac{1}{8}z&=2\\end{align}[\/latex]<\/p>\n
33. [latex]\\begin{align}\\frac{4}{5}x-\\frac{7}{8}y+\\frac{1}{2}z&=1 \\\\ -\\frac{4}{5}x-\\frac{3}{4}y+\\frac{1}{3}z&=-8 \\\\ -\\frac{2}{5}x-\\frac{7}{8}y+\\frac{1}{2}z&=-5 \\end{align}[\/latex]<\/p>\n
34.\u00a0[latex]\\begin{align} -\\frac{1}{3}x-\\frac{1}{8}y+\\frac{1}{6}z&=-\\frac{4}{3} \\\\ -\\frac{2}{3}x-\\frac{7}{8}y+\\frac{1}{3}z&=-\\frac{23}{3} \\\\ -\\frac{1}{3}x-\\frac{5}{8}y+\\frac{5}{6}z&=0 \\end{align}[\/latex]<\/p>\n
35. [latex]\\begin{align} -\\frac{1}{4}x-\\frac{5}{4}y+\\frac{5}{2}z&=-5 \\\\ -\\frac{1}{2}x-\\frac{5}{3}y+\\frac{5}{4}z&=\\frac{55}{12} \\\\ -\\frac{1}{3}x-\\frac{1}{3}y+\\frac{1}{3}z&=\\frac{5}{3} \\end{align}[\/latex]<\/p>\n
36.\u00a0[latex]\\begin{align}\\frac{1}{40}x+\\frac{1}{60}y+\\frac{1}{80}z&=\\frac{1}{100} \\\\ -\\frac{1}{2}x-\\frac{1}{3}y-\\frac{1}{4}z&=-\\frac{1}{5} \\\\ \\frac{3}{8}x+\\frac{3}{12}y+\\frac{3}{16}z&=\\frac{3}{20} \\end{align}[\/latex]<\/p>\n
37. [latex]\\begin{align}0.1x - 0.2y+0.3z&=2\\\\ 0.5x - 0.1y+0.4z&=8\\\\ 0.7x - 0.2y+0.3z&=8\\end{align}[\/latex]<\/p>\n
38.\u00a0[latex]\\begin{align}0.2x+0.1y - 0.3z&=0.2\\\\ 0.8x+0.4y - 1.2z&=0.1\\\\ 1.6x+0.8y - 2.4z&=0.2\\end{align}[\/latex]<\/p>\n
39. [latex]\\begin{align}1.1x+0.7y - 3.1z&=-1.79\\\\ 2.1x+0.5y - 1.6z&=-0.13\\\\ 0.5x+0.4y - 0.5z&=-0.07\\end{align}[\/latex]<\/p>\n
40.\u00a0[latex]\\begin{align}0.5x - 0.5y+0.5z&=10\\\\ 0.2x - 0.2y+0.2z&=4\\\\ 0.1x - 0.1y+0.1z&=2\\end{align}[\/latex]<\/p>\n
41. [latex]\\begin{align}0.1x+0.2y+0.3z&=0.37\\\\ 0.1x - 0.2y - 0.3z&=-0.27\\\\ 0.5x - 0.1y - 0.3z&=-0.03\\end{align}[\/latex]<\/p>\n
42.\u00a0[latex]\\begin{align}0.5x - 0.5y - 0.3z&=0.13\\\\ 0.4x - 0.1y - 0.3z&=0.11\\\\ 0.2x - 0.8y - 0.9z&=-0.32\\end{align}[\/latex]<\/p>\n
43. [latex]\\begin{align}0.5x+0.2y - 0.3z&=1\\\\ 0.4x - 0.6y+0.7z&=0.8\\\\ 0.3x - 0.1y - 0.9z&=0.6\\end{align}[\/latex]<\/p>\n
44.\u00a0[latex]\\begin{align}0.3x+0.3y+0.5z&=0.6\\\\ 0.4x+0.4y+0.4z&=1.8\\\\ 0.4x+0.2y+0.1z&=1.6\\end{align}[\/latex]<\/p>\n
45. [latex]\\begin{align}0.8x+0.8y+0.8z&=2.4\\\\ 0.3x - 0.5y+0.2z&=0\\\\ 0.1x+0.2y+0.3z&=0.6\\end{align}[\/latex]<\/p>\n
For the following exercises, solve the system for [latex]x,y[\/latex], and [latex]z[\/latex].<\/p>\n
46. [latex]\\begin{align}x+y+z&=3 \\\\ \\frac{x - 1}{2}+\\frac{y - 3}{2}+\\frac{z+1}{2}&=0 \\\\ \\frac{x - 2}{3}+\\frac{y+4}{3}+\\frac{z - 3}{3}&=\\frac{2}{3} \\end{align}[\/latex]<\/p>\n
47. [latex]\\begin{align}5x - 3y-\\frac{z+1}{2}&=\\frac{1}{2} \\\\ 6x+\\frac{y - 9}{2}+2z&=-3 \\\\ \\frac{x+8}{2}-4y+z&=4 \\end{align}[\/latex]<\/p>\n
48.\u00a0[latex]\\begin{align}\\frac{x+4}{7}-\\frac{y - 1}{6}+\\frac{z+2}{3}&=1\\\\ \\frac{x - 2}{4}+\\frac{y+1}{8}-\\frac{z+8}{12}&=0\\\\ \\frac{x+6}{3}-\\frac{y+2}{3}+\\frac{z+4}{2}7&=3\\end{align}[\/latex]<\/p>\n
49. [latex]\\begin{align}\\frac{x - 3}{6}+\\frac{y+2}{2}-\\frac{z - 3}{3}&=2\\\\ \\frac{x+2}{4}+\\frac{y - 5}{2}+\\frac{z+4}{2}&=1\\\\ \\frac{x+6}{2}-\\frac{y - 3}{2}+z+1&=9\\end{align}[\/latex]<\/p>\n
50.\u00a0[latex]\\begin{align}\\frac{x - 1}{3}+\\frac{y+3}{4}+\\frac{z+2}{6}&=1 \\\\ 4x+3y - 2z&=11 \\\\ 0.02x+0.015y - 0.01z&=0.065 \\end{align}[\/latex]<\/p>\n\n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t