Contoh Soal 3 Sistem Persamaan Linear Tiga Variabel

$\begin{array}{l}\\ 11.& \text{Suatu bilangan terdiri atas 3 angka. Jumlah}\\ & \text{ketiga angka tersebut adalah 9. Angka kedua}\\ & \text{dikurangi angka pertama dan angka ketiga }\\ & \text{sama dengan 1. Dua kali angka pertama sama}\\ & \text{dengan jumlah angka kedua dan angka ketiga.}\\ & \text{Angka puluhan pada bilangan tersebut adalah}\\ & ....\\ & \begin{array}{l}\\ \text{a}.& 3\\ \text{b}.& 4\\ \text{c}.& 5\\ \text{d}.& 6\\ \text{e}.& 7\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \text{Model matematikanya}\\ & \{\begin{array}{c}A+B+C=9....(1)\\ 2B-A-C=1....(2)\\ 2A=B+C....(3)\end{array}\\ & \text{Saat}(1)+(2),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle A+B+C}& =9\\ {\displaystyle -A+B-C}& =1& +\\ 2B& =10\\ B& =5& ...(4)\end{array}\\ & \text{Jadi},\text{bilangan kedua adalah}=B=5\end{array}\end{array}$

$\begin{array}{l}\\ 12.& (\mathbf{\text{SIMAK UI 2010}})\\ & \text{Jika}x+y+2z=K,x+2y+z=K,\\ & 2x+y+z=K\text{dengan}K\ne 0,\text{maka}\\ & x^2+y^2+z^2\text{bila dinyatakan dalam}K\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{1}{16}{K}^2}\\ \text{b}.& {\displaystyle \frac{3}{16}{K}^2}\\ \text{c}.& {\displaystyle \frac{4}{17}{K}^2}\\ \text{d}.& {\displaystyle \frac{3}{8}{K}^2}\\ \text{e}.& {\displaystyle \frac{2}{3}{K}^2}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}x+y+2z=K....(1)\\ x+2y+z=K....(2)\\ 2x+y+z=K....(3)\end{array}\\ & \text{maka}\\ & \{\begin{array}{c}z+(x+y+z)=K....(1)\\ y+(x+y+z)=K....(2)\\ x+(x+y+z)=K....(3)\end{array}\\ & \text{Saat}(1)+(2)+(3),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle x+y+2z}& =K\\ {\displaystyle x+2y+z}& =K& \\ {\displaystyle 2x+y+z}& =K& +\\ 4x+4y+4z& =3K\\ x+y+z& ={\displaystyle \frac{3}{4}K}& ...(4)\end{array}\\ & \text{Saat}(4)\text{disubstitusikan ke}(1),(2),\text{dan}(3)\\ & \text{Jelas bahwa akan didapatkan}\\ & x=y=z={\displaystyle \frac{1}{4}K}\\ & \text{Jadi},x^2+y^2+y^2=3{\left({\displaystyle \frac{1}{4}K}\right)}^2={\displaystyle \frac{3}{16}{K}^2}\end{array}\end{array}$

$\begin{array}{l}\\ 13.& \text{Diketahui}0,15252525252...={\displaystyle \frac{p}{2q+r}}\\ & \text{Jika jumlah}p\text{dan}q=\text{3 kali}r,\text{maka}\\ & \text{masing-masing harga}p,q,\text{dan}r=....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 152,2819,2584}\\ \text{b}.& {\displaystyle 252,{\displaystyle \frac{5638}{7},\frac{8102}{21}}}\\ \text{c}.& {\displaystyle 151,\frac{2819}{7},\frac{1292}{7}}\\ \text{d}.& {\displaystyle 151,{\displaystyle \frac{2819}{7},\frac{2584}{7}}}\\ \text{e}.& {\displaystyle 152,\frac{2819}{14},\frac{1292}{7}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \text{Dari soal diketahui}\\ & \{\begin{array}{ll}0,1\overline{5252}& ={\displaystyle \frac{p}{2q+r}.....(1)}\\ p+q& =3r............(2)\end{array}\\ & \text{dan}\\ & \begin{array}{l}\\ {\displaystyle x}& =0,15252525252...\\ {\displaystyle 1000x}& =152,5252525252...\\ {\displaystyle 10x}& =1,5252525252...& -\\ 990x& =151\\ x& ={\displaystyle \frac{151}{990},\text{maka}}\\ {\displaystyle \frac{p}{2q+r}}& ={\displaystyle \frac{151}{990}}\\ & \{\begin{array}{ll}p& =151.......(3)\\ 2p+r& =990.......(4)\end{array}\end{array}\\ & \text{Dari}(3)\text{diperoleh}:q=3r-p=3r-151....(5)\\ & \text{Dari}(5)\text{disubstitusikan ke}(4)\\ & \begin{array}{rl}2q+r& =990\\ 2(3r-151)+r& =990\\ 6r-302+r& =990\\ 7r& =990+302=1292\\ r& ={\displaystyle \frac{1292}{7}.....(6)}\end{array}\\ & \text{Dari}(3)\mathrm{\&}(6)\text{disubstitusikan ke}(2)\\ & \begin{array}{rl}p+q& =3r\\ 151+q& =3\left({\displaystyle \frac{1292}{7}}\right)\\ q& ={\displaystyle \frac{3876}{7}-151}\\ & ={\displaystyle \frac{3876-1057}{7}}\\ & ={\displaystyle \frac{2819}{7}.....(7)}\end{array}\\ & \text{Jadi},p,q,r\text{adalah}:{\displaystyle 151,\frac{2819}{7},\frac{1292}{7}}\end{array}\end{array}$

$\begin{array}{l}\\ 14.& \text{Perhatikanlah sistem persamaan berikut}\\ & \{\begin{array}{ll}3x+2y-5z& =3\\ 2x-6y+kz& =9\\ 5x-4y-z& =5\end{array}\\ & \text{agar sistem persamaan ini tidak}\\ & \text{memiliki penyelesaian, maka nilai}k=....\\ & \begin{array}{l}\\ \text{a}.& -4\\ \text{b}.& 2\\ \text{c}.& 3\\ \text{d}.& 4\\ \text{e}.& 6\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}& \text{Agar sistem persamaan}\\ & \{\begin{array}{ll}3x+2y-5z& =3\\ 2x-6y+kz& =9\\ 5x-4y-z& =5\end{array}\\ & \text{tidak berpenyelesaian, maka}\\ & \text{ingat penyelesaian metode matrik}\\ & \text{buatlah penyebutnya}=0,\text{yaitu}:\\ & \left|\begin{array}{ccc}3& 2& -5\\ 2& -6& k\\ 5& -4& -1\end{array}\right|=0\\ & \text{Selanjutnya}\\ & 3\left|\begin{array}{cc}-6& k\\ -4& -1\end{array}\right|-2\left|\begin{array}{cc}2& k\\ 5& -1\end{array}\right|-5\left|\begin{array}{cc}2& -6\\ 5& -4\end{array}\right|=0\\ & 3(6+4k)-2(-2-5k)-5(-8+30)=0\\ & 18+12k+4+10k+40-150=0\\ & 22x=88\\ & x=4\end{array}\end{array}$

$\begin{array}{l}\\ 15.& \text{Diketahui}\\ & \left(\begin{array}{ccc}{\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{5}}\\ {\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{5}}& -{\displaystyle \frac{4}{5}}\\ -{\displaystyle \frac{2}{5}}& {\displaystyle \frac{1}{10}}& {\displaystyle \frac{1}{10}}\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}1\\ 2\\ 0\end{array}\right)\\ & \text{Nilai}x,y,\text{dan}z\text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle \frac{1}{5},\frac{4}{5},-\frac{1}{10}}\\ \text{b}.& -1,5,1\\ \text{c}.& 1,5,-1\\ \text{d}.& -1,1,5\\ \text{e}.& 5,1,-1\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \text{Diketahui sistem persamaan}\\ & \{\begin{array}{c}{\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z=1....(1)}\\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y-\frac{4}{5}z=2....(2)}\\ -{\displaystyle \frac{2}{5}x+\frac{1}{10}y+\frac{1}{10}z=0....(3)}\end{array}\\ & \text{Saat}(1)+(2),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z}& =1& \\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y-\frac{4}{5}z}& =2& -\\ {\displaystyle \frac{5}{5}z}& =-1& \\ {\displaystyle z}& =-1& ...(4)\end{array}\\ & \text{Saat}(1)+(3),\text{maka}\\ & \begin{array}{l}\\ {\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z}& =1& |\times 1|& {\displaystyle \frac{1}{5}x+\frac{1}{5}y+\frac{1}{5}z=1}\\ -{\displaystyle \frac{2}{5}x+\frac{1}{10}y+\frac{1}{10}z}& =0& |\times 2|& -{\displaystyle \frac{4}{5}+\frac{1}{5}y+\frac{1}{5}z=0}& -\\ & & & {\displaystyle \frac{5}{5}x=1}& \\ & & & x=-1........(5)\end{array}\\ & \text{Dari persamaan}(4)\mathrm{\&}(5)\text{akan didapatkan}\\ & y=5\\ & \text{Jadi},(x,y,z)=(1,5,-1)\end{array}\end{array}$