11.2 Systems of linear equations: three variables  (Page 5/8)

$\begin{array}{l}2x\mathrm{-6}y+6z=\mathrm{-12}\hfill \\ x+4y+5z=\mathrm{-1}\hfill \\ \mathrm{-x}+2y+3z=\mathrm{-1}\hfill \end{array}$ and $(0,1,\mathrm{-1})$

$\begin{array}{l}6x-y+3z=6\hfill \\ 3x+5y+2z=0\hfill \\ x+y=0\hfill \end{array}$ and $(3,\mathrm{-3},\mathrm{-5})$

$\begin{array}{l}6x\mathrm{-7}y+z=2\hfill \\ \mathrm{-x}-y+3z=4\hfill \\ 2x+y-z=1\hfill \end{array}$ and $(4,2,\mathrm{-6})$

$\begin{array}{l}x-y=0\hfill \\ x-z=5\hfill \\ x-y+z=\mathrm{-1}\hfill \end{array}$ and $(4,4,\mathrm{-1})$

$\begin{array}{l} \mathrm{-x}-y+2z=3\hfill \\ 5x+8y\mathrm{-3}z=4\hfill \\ \mathrm{-x}+3y\mathrm{-5}z=\mathrm{-5}\hfill \end{array}$ and $(4,1,\mathrm{-7})$

$\begin{array}{l}3x\mathrm{-4}y+2z=\mathrm{-15}\hfill \\ 2x+4y+z=16\hfill \\ 2x+3y+5z=20\hfill \end{array}$

$\begin{array}{l}5x\mathrm{-2}y+3z=20\hfill \\ 2x\mathrm{-4}y\mathrm{-3}z=\mathrm{-9}\hfill \\ x+6y\mathrm{-8}z=21\hfill \end{array}$

$\begin{array}{l}5x+2y+4z=9\hfill \\ \mathrm{-3}x+2y+z=10\hfill \\ 4x\mathrm{-3}y+5z=\mathrm{-3}\hfill \end{array}$

$\begin{array}{l}4x\mathrm{-3}y+5z=31\hfill \\ -x+2y+4z=20\hfill \\ x+5y\mathrm{-2}z=\mathrm{-29}\hfill \end{array}$

$\begin{array}{l}5x\mathrm{-2}y+3z=4\hfill \\ \mathrm{-4}x+6y\mathrm{-7}z=\mathrm{-1}\hfill \\ 3x+2y-z=4\hfill \end{array}$

$\begin{array}{l} 4x+6y+9z=0\hfill \\ \mathrm{-5}x+2y\mathrm{-6}z=3\hfill \\ 7x\mathrm{-4}y+3z=\mathrm{-3}\hfill \end{array}$

$\begin{array}{l}2x-y+3z=17\hfill \\ \mathrm{-5}x+4y\mathrm{-2}z=\mathrm{-46}\hfill \\ 2y+5z=\mathrm{-7}\hfill \end{array}$

$\begin{array}{l}5x\mathrm{-6}y+3z=50\hfill \\ -x+4y=10\hfill \\ 2x-z=10\hfill \end{array}$

$\begin{array}{l}2x+3y\mathrm{-6}z=1\hfill \\ \mathrm{-4}x\mathrm{-6}y+12z=\mathrm{-2}\hfill \\ x+2y+5z=10\hfill \end{array}$

$\begin{array}{l}4x+6y\mathrm{-2}z=8\hfill \\ 6x+9y\mathrm{-3}z=12\hfill \\ \mathrm{-2}x\mathrm{-3}y+z=\mathrm{-4}\hfill \end{array}$

$\begin{array}{l}2x+3y\mathrm{-4}z=5\hfill \\ \mathrm{-3}x+2y+z=11\hfill \\ -x+5y+3z=4\hfill \end{array}$

$\begin{array}{l}10x+2y\mathrm{-14}z=8\hfill \\ \mathrm{-x}\mathrm{-2}y\mathrm{-4}z=\mathrm{-1}\hfill \\ \mathrm{-12}x\mathrm{-6}y+6z=\mathrm{-12}\hfill \end{array}$

$\begin{array}{l}x+y+z=14\hfill \\ 2y+3z=\mathrm{-14}\hfill \\ \mathrm{-16}y\mathrm{-24}z=\mathrm{-112}\hfill \end{array}$

$\begin{array}{l}5x\mathrm{-3}y+4z=\mathrm{-1}\hfill \\ \mathrm{-4}x+2y\mathrm{-3}z=0\hfill \\ \mathrm{-x}+5y+7z=\mathrm{-11}\hfill \end{array}$

$\begin{array}{l}x+y+z=0\hfill \\ 2x-y+3z=0\hfill \\ x-z=0\hfill \end{array}$

$\begin{array}{l}3x+2y\mathrm{-5}z=6\\ 5x\mathrm{-4}y+3z=\mathrm{-12}\\ 4x+5y\mathrm{-2}z=15\end{array}$

$\begin{array}{l}x+y+z=0\hfill \\ 2x-y+3z=0\hfill \\ x-z=1\hfill \end{array}$

$\begin{array}{l}\begin{array}{l}\\ 3x-\frac{1}{2}y-z=-\frac{1}{2}\end{array}\hfill \\ 4x+z=3\hfill \\ -x+\frac{3}{2}y=\frac{5}{2}\hfill \end{array}$

$\begin{array}{l}6x\mathrm{-5}y+6z=38\hfill \\ \frac{1}{5}x-\frac{1}{2}y+\frac{3}{5}z=1\hfill \\ \mathrm{-4}x-\frac{3}{2}y-z=\mathrm{-74}\hfill \end{array}$

$\begin{array}{l}\frac{1}{2}x-\frac{1}{5}y+\frac{2}{5}z=-\frac{13}{10}\hfill \\ \frac{1}{4}x-\frac{2}{5}y-\frac{1}{5}z=-\frac{7}{20}\hfill \\ -\frac{1}{2}x-\frac{3}{4}y-\frac{1}{2}z=-\frac{5}{4}\hfill \end{array}$

$\begin{array}{l}\begin{array}{l}\\ -\frac{1}{3}x-\frac{1}{2}y-\frac{1}{4}z=\frac{3}{4}\end{array}\hfill \\ -\frac{1}{2}x-\frac{1}{4}y-\frac{1}{2}z=2\hfill \\ -\frac{1}{4}x-\frac{3}{4}y-\frac{1}{2}z=-\frac{1}{2}\hfill \end{array}$

$\begin{array}{l}\frac{1}{2}x-\frac{1}{4}y+\frac{3}{4}z=0\\ \frac{1}{4}x-\frac{1}{10}y+\frac{2}{5}z=\mathrm{-2}\\ \frac{1}{8}x+\frac{1}{5}y-\frac{1}{8}z=2\end{array}$

$\begin{array}{l}\frac{4}{5}x-\frac{7}{8}y+\frac{1}{2}z=1\hfill \\ -\frac{4}{5}x-\frac{3}{4}y+\frac{1}{3}z=\mathrm{-8}\hfill \\ -\frac{2}{5}x-\frac{7}{8}y+\frac{1}{2}z=\mathrm{-5}\hfill \end{array}$

$\begin{array}{l}\begin{array}{l}\\ -\frac{1}{3}x-\frac{1}{8}y+\frac{1}{6}z=-\frac{4}{3}\end{array}\hfill \\ -\frac{2}{3}x-\frac{7}{8}y+\frac{1}{3}z=-\frac{23}{3}\hfill \\ -\frac{1}{3}x-\frac{5}{8}y+\frac{5}{6}z=0\hfill \end{array}$

$\begin{array}{l}\begin{array}{l}\\ -\frac{1}{4}x-\frac{5}{4}y+\frac{5}{2}z=\mathrm{-5}\end{array}\hfill \\ -\frac{1}{2}x-\frac{5}{3}y+\frac{5}{4}z=\frac{55}{12}\hfill \\ -\frac{1}{3}x-\frac{1}{3}y+\frac{1}{3}z=\frac{5}{3}\hfill \end{array}$

$\begin{array}{l}\frac{1}{40}x+\frac{1}{60}y+\frac{1}{80}z=\frac{1}{100}\hfill \\ -\frac{1}{2}x-\frac{1}{3}y-\frac{1}{4}z=-\frac{1}{5}\hfill \\ \frac{3}{8}x+\frac{3}{12}y+\frac{3}{16}z=\frac{3}{20}\hfill \end{array}$

$\begin{array}{l}0.1x\mathrm{-0.2}y+0.3z=2\\ 0.5x\mathrm{-0.1}y+0.4z=8\\ 0.7x\mathrm{-0.2}y+0.3z=8\end{array}$

$\begin{array}{l}0.2x+0.1y\mathrm{-0.3}z=0.2\\ 0.8x+0.4y\mathrm{-1.2}z=0.1\\ 1.6x+0.8y\mathrm{-2.4}z=0.2\end{array}$

$\begin{array}{l}1.1x+0.7y\mathrm{-3.1}z=\mathrm{-1.79}\\ 2.1x+0.5y\mathrm{-1.6}z=\mathrm{-0.13}\\ 0.5x+0.4y\mathrm{-0.5}z=\mathrm{-0.07}\end{array}$

$\begin{array}{l}0.5x\mathrm{-0.5}y+0.5z=10\\ 0.2x\mathrm{-0.2}y+0.2z=4\\ 0.1x\mathrm{-0.1}y+0.1z=2\end{array}$

$\begin{array}{l}0.1x+0.2y+0.3z=0.37\\ 0.1x\mathrm{-0.2}y\mathrm{-0.3}z=\mathrm{-0.27}\\ 0.5x\mathrm{-0.1}y\mathrm{-0.3}z=\mathrm{-0.03}\end{array}$

$\begin{array}{l}0.5x\mathrm{-0.5}y\mathrm{-0.3}z=0.13\\ 0.4x\mathrm{-0.1}y\mathrm{-0.3}z=0.11\\ 0.2x\mathrm{-0.8}y\mathrm{-0.9}z=\mathrm{-0.32}\end{array}$

$\begin{array}{l}0.5x+0.2y\mathrm{-0.3}z=1\\ 0.4x\mathrm{-0.6}y+0.7z=0.8\\ 0.3x\mathrm{-0.1}y\mathrm{-0.9}z=0.6\end{array}$

$\begin{array}{l}0.3x+0.3y+0.5z=0.6\\ 0.4x+0.4y+0.4z=1.8\\ 0.4x+0.2y+0.1z=1.6\end{array}$

$\begin{array}{l}0.8x+0.8y+0.8z=2.4\\ 0.3x\mathrm{-0.5}y+0.2z=0\\ 0.1x+0.2y+0.3z=0.6\end{array}$

$\begin{array}{l}x+y+z=3\hfill \\ \frac{x\mathrm{-1}}{2}+\frac{y\mathrm{-3}}{2}+\frac{z+1}{2}=0\hfill \\ \frac{x\mathrm{-2}}{3}+\frac{y+4}{3}+\frac{z\mathrm{-3}}{3}=\frac{2}{3}\hfill \end{array}$

$\begin{array}{l}5x\mathrm{-3}y-\frac{z+1}{2}=\frac{1}{2}\hfill \\ 6x+\frac{y\mathrm{-9}}{2}+2z=\mathrm{-3}\hfill \\ \frac{x+8}{2}\mathrm{-4}y+z=4\hfill \end{array}$

$\begin{array}{l}\frac{x+4}{7}-\frac{y\mathrm{-1}}{6}+\frac{z+2}{3}=1\\ \frac{x\mathrm{-2}}{4}+\frac{y+1}{8}-\frac{z+8}{12}=0\\ \frac{x+6}{3}-\frac{y+2}{3}+\frac{z+4}{2}=3\end{array}$

$\begin{array}{l}\frac{x\mathrm{-3}}{6}+\frac{y+2}{2}-\frac{z\mathrm{-3}}{3}=2\\ \frac{x+2}{4}+\frac{y\mathrm{-5}}{2}+\frac{z+4}{2}=1\\ \frac{x+6}{2}-\frac{y\mathrm{-3}}{2}+z+1=9\end{array}$

$\begin{array}{l}\frac{x\mathrm{-1}}{3}+\frac{y+3}{4}+\frac{z+2}{6}=1\hfill \\ 4x+3y\mathrm{-2}z=11\hfill \\ 0.02x+0.015y\mathrm{-0.01}z=0.065\hfill \end{array}$

Three even numbers sum up to 108. The smaller is half the larger and the middle number is $\frac{3}{4}$ the larger. What are the three numbers?

Three numbers sum up to 147. The smallest number is half the middle number, which is half the largest number. What are the three numbers?

At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?