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Senin, 06 Mei 2019

Sistem Persamaan Linear Tigaa Variabel (SPLTV)

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Sistem Persamaan Linear Tigaa Variabel (SPLTV)
\begin{array}{ll}\\ \underline{\textbf{Bentuk Umum}}&:\\ &\begin{cases} a_{1}x+b_{1}y+c_{1}z=d_{1} \\ a_{2}x+b_{2}y+c_{2}z=d_{2} \\ a_{3}x+b_{3}y+c_{3}z=d_{3} \end{cases}\\\\ \qquad \quad \textbf{dengan}&\bullet \quad a_{1},\: a_{2},\: a_{3},\\ &\, \: \: \quad b_{1},\: b_{2},\: b_{3},\\ &\, \: \: \quad c_{1},\: c_{2},\: c_{3},\\ &\, \: \: \quad d_{1},\: d_{2},\: d_{3}\\ &\: \: \quad \textrm{semuanya adalah bilangan real} \end{array}

Menyelesaikan Sistem Persamaan linear dua dan tiga variabel


by 
Perhatikan SPLDV dan SPLTV berikut
\LARGE \begin{array}{lllll}\\ \begin{aligned}2x-y&=-7\\ 5x-6y&=-28\\ & \end{aligned}&&\textrm{dan}&&\begin{aligned}x-2y+2z&=3\\ 2x-y-3z&=-9\\ 3x+2y-z&=4 \end{aligned} \end{array}.
Permasalahan di atas dapat kita selesaikan dengan konsep matriks. Lihat kembali bagian Determinan dan Invers matriks.
Penyelesaian dengan konsep matriks adalah sebagai berikut
  • dengan menerapkan metode Invers, atau
  • dengan menggunakan determinan yang selanjutnya lebih dikenal dengan aturan Cramer
\begin{array}{|c|c|c|}\hline \textrm{Metode}&\textbf{SPLDV}&\textbf{SPLTV}\\\hline \textrm{Invers}&\begin{aligned}ax+by&=p\\ cx+dy&=q\\ & \end{aligned}&\begin{aligned}ax+by+cz&=r\\ dx+ey+fz&=s\\ gx+hy+iz&=t \end{aligned}\\\cline{2-3} &\begin{aligned}\begin{pmatrix} a & b\\ c & d \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}&=\begin{pmatrix} p\\ q \end{pmatrix}\\ \begin{pmatrix} x\\ y \end{pmatrix}&=\begin{pmatrix} a & b\\ c & d \end{pmatrix}^{-1}\begin{pmatrix} p\\ q \end{pmatrix}\\ &\\ & \end{aligned}&\begin{aligned}\begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}\begin{pmatrix} x\\ y\\ z \end{pmatrix}&=\begin{pmatrix} r\\ s\\ t \end{pmatrix}\\ \begin{pmatrix} x\\ y\\ z \end{pmatrix}&=\begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}^{-1}\begin{pmatrix} r\\ s\\ t \end{pmatrix} \end{aligned} \\\hline \textrm{Determinan}&\begin{aligned}x&=\displaystyle \frac{\begin{vmatrix} p & b\\ q & d \end{vmatrix}}{\begin{vmatrix} a & b\\ c & d \end{vmatrix}}\\ &\textrm{dan}\\ y&=\displaystyle \frac{\begin{vmatrix} a & p\\ c & q \end{vmatrix}}{\begin{vmatrix} a & b\\ c & d \end{vmatrix}}\\ &\\ &\\ &\\ &\\ &\\ &\\ &\\ &\\ & \end{aligned}&\begin{aligned}x&=\displaystyle \frac{\begin{vmatrix} r & b & c\\ s & e & f\\ t & h & i \end{vmatrix}}{\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}}\\ &\textrm{dan}\\ y&=\displaystyle \frac{\begin{vmatrix} a & r & c\\ d & s & f\\ g & t & i \end{vmatrix}}{\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}}\\ &\textrm{serta}\\ z&=\displaystyle \frac{\begin{vmatrix} a & b & r\\ d & e & s\\ g & h & t \end{vmatrix}}{\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}} \end{aligned}\\\hline \end{array}.
\LARGE{\fbox{\LARGE{\fbox{Contoh Soal}}}}.
\begin{array}{|l|l|}\hline \textrm{SPLDV}&\textrm{SPLTV}\\\hline \begin{aligned}2x-y&=-7\\ 5x-6y&=-28\\ & \end{aligned}&\begin{aligned}x-2y+2z&=3\\ 2x-y-3z&=-9\\ 3x+2y-z&=4 \end{aligned}\\\hline \multicolumn{2}{|c|}{\textrm{Penyelesaian}}\\\hline \begin{aligned}x&=\displaystyle \frac{\begin{vmatrix} p & b\\ q & d \end{vmatrix}}{\begin{vmatrix} a & b\\ c & d \end{vmatrix}}=\displaystyle \frac{\begin{vmatrix} -7 & -1\\ -28 & -6 \end{vmatrix}}{\begin{vmatrix} 2 & -1\\ 5 & -6 \end{vmatrix}}\\ &=\displaystyle \frac{(42-28)}{(-12-(-5))}=\displaystyle \frac{14}{-7}\\ &=-2\\ \textrm{dan}&\\ y&=\displaystyle \frac{\begin{vmatrix} a & p\\ c & q \end{vmatrix}}{\begin{vmatrix} a & b\\ c & d \end{vmatrix}}=\displaystyle \frac{\begin{vmatrix} 2 & -7\\ 5 & -28 \end{vmatrix}}{\begin{vmatrix} 2 & -1\\ 5 & -6 \end{vmatrix}}\\ &=\displaystyle \frac{(-56)-(-35)}{(-12)-(-5)}\\ &=\displaystyle \frac{-21}{-7}\\ &=3 \end{aligned}&\begin{aligned}\textrm{Silahkan dicoba sendiri} \end{aligned}\\\hline \end{array}.

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