Exercise Zone : Sistem Persamaan Linear Tiga Variabel

Berikut ini adalah kumpulan soal mengenai Sistem Persamaan Linear Tiga Variabel tipe standar. Jika ingin bertanya soal, silahkan gabung ke grup Facebook atau Telegram.

Tipe:

Standar SBMPTN

No. 1

{a,b,c} adalah himpunan penyelesaian $$\{\begin{array}{l}3a+2b+c=7\\ a+b+2c=7\\ a+2b+2c=6\end{array}$$ Nilai a+b+c adalah

1. -2
2. 0
3. 2

4. 4
5. 6

$\begin{array}{rl}3a+2b+c& =7\\ a+b+2c& =7\\ a+2b+2c& =6& +\\ 5a+5b+5c& =20\\ a+b+c& ={\displaystyle \frac{20}{5}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 4}}}}\end{array}$

Jadi, a+b+c=4.
JAWAB: D

No. 2

Diketahui sistem persamaan linear berikut $$\{\begin{array}{ll}ax+y+z& =1\\ x+ay+z& =a\\ x+y+az& =a^2\end{array}$$

1. Tentukan penyelesaian sistem persamaan linear tersebut dinyatakan dalam a.
   Penuntun: jumlahkan ketiga persamaan dan carilah nilai x, y, dan z
2. Tentukan nilai a agar sistem tersebut tidak mempunyai penyelesaian.
   Penuntun: libatkan pembagian dengan nol.
3. Apakah ada nilai a yang menyebabkan sistem tersebut mempunyai banyak penyelesaian?

Grup Whatsapp

1. 
   $\begin{array}{rl}ax+y+z& =1\\ x+ay+z& =a\\ x+y+az& =a^2& +\\ (a+2)x+(a+2)y+(a+2)z& =a^2+a+1\\ (a+2)(x+y+z)& =a^2+a+1\\ x+y+z& ={\displaystyle \frac{a^2+a+1}{a+2}}\end{array}$
   
   $\begin{array}{rl}ax+y+z& =1\\ ax+{\displaystyle \frac{a^2+a+1}{a+2}}-x& =1\\ (a-1)x& =1-{\displaystyle \frac{a^2+a+1}{a+2}}\\ & ={\displaystyle \frac{a+2-(a^2+a+1)}{a+2}}\\ & ={\displaystyle \frac{a+2-a^2-a-1}{a+2}}\\ & ={\displaystyle \frac{-a^2+1}{a+2}}\\ x& ={\displaystyle \frac{-(a+1)(a-1)}{(a-1)(a+2)}}\\ & ={\displaystyle \frac{-a-1}{a+2}}\end{array}$
   
   $\begin{array}{rl}x+ay+z& =a\\ ay+{\displaystyle \frac{a^2+a+1}{a+2}}-y& =a\\ (a-1)y& =a-{\displaystyle \frac{a^2+a+1}{a+2}}\\ & ={\displaystyle \frac{a(a+2)-(a^2+a+1)}{a+2}}\\ & ={\displaystyle \frac{a^2+2a-a^2-a-1}{a+2}}\\ y& ={\displaystyle \frac{a-1}{(a-1)(a+2)}}\\ & ={\displaystyle \frac{1}{a+2}}\end{array}$
   
   $\begin{array}{rl}x+y+az& =a^2\\ az+{\displaystyle \frac{a^2+a+1}{a+2}}-z& =a^2\\ (a-1)z& =a^2-{\displaystyle \frac{a^2+a+1}{a+2}}\\ & ={\displaystyle \frac{a^2(a+2)-(a^2+a+1)}{a+2}}\\ & ={\displaystyle \frac{a^3+2a^2-a^2-a-1}{a+2}}\\ z& ={\displaystyle \frac{a^3+a^2-a-1}{(a-1)(a+2)}}\\ & ={\displaystyle \frac{(a-1)(a^2+2a+1)}{(a-1)(a+2)}}\\ & ={\displaystyle \frac{a^2+2a+1}{a+2}}\end{array}$
   
2. agar tidak mempunyai penyelesaian, maka:
   $\begin{array}{rl}a+2& =0\\ a& =-2\end{array}$
   
3. tidak ada

Jadi,

1. x=\dfrac{-a-1}{a+2}, y=\dfrac1{a+2}, z=\dfrac{a^2+2a+1}{a+2}.
2. a=-2
3. tidak ada

No. 3

Himpunan penyelesaian dari SPLTV $$\{\begin{array}{ll}x+y-z& =-1\\ 2x-2y+3z& =8\\ 2x-y+2z& =9\end{array}$$ adalah ....

1. (-15,2,14)
2. (-14,2,15)
3. (-2,-15,14)

4. (-2,15,14)
5. (-2,14,15)

Grup Telegram

CARA BIASA

(1) dan (2)

x+y-z=-1\ {\color{red}\times2}

$\begin{array}{rl}2x+2y-2z& =-2\\ 2x-2y+3z& =8& -\\ 4y-5z& =-10& (4)\end{array}$

(2) dan (3)

$\begin{array}{rl}2x-2y+3z& =8\\ 2x-y+2z& =9& -\\ -y+z& =-1& (5)\end{array}$

(4) dan (5)

$\begin{array}{rl}4y-5z& =-10\\ -y+z& =-1& \times 4\end{array}$

$\begin{array}{rl}4y-5z& =-10\\ -4y+4z& =-4& +\\ -z& =-14\\ z& =\boxed{{\displaystyle \boxed{{\displaystyle 14}}}}\end{array}$

Substitusikan ke (5)
$\begin{array}{rl}-y+z& =-1\\ -y+14& =-1\\ -y& =-15\\ y& =\boxed{{\displaystyle \boxed{{\displaystyle 15}}}}\end{array}$

Substitusikan ke (1)
$\begin{array}{rl}x+y-z& =-1\\ x+15-14& =-1\\ x+1& =-1\\ x& =\boxed{{\displaystyle \boxed{{\displaystyle -2}}}}\end{array}$

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CARA CEPAT

{2x-y+2z=9\ {\color{red}\times-1}\rightarrow -2x+y-2z=-9}

$\begin{array}{rl}x+y-z& =-1\\ 2x-2y+3z& =8\\ -2x+y-2z& =-9& +\\ x& =\boxed{{\displaystyle \boxed{{\displaystyle -2}}}}\end{array}$

$\begin{array}{rll}x+y-z& =-1& \times -2\\ 2x-2y+3z& =8& \times -3\\ 2x-y+2z& =9& \times 4\end{array}$

$\begin{array}{rl}-2x-2y+2z& =2\\ -6x+6y-9z& =-24\\ 8x-4y+8z& =36& +\\ z& =\boxed{{\displaystyle \boxed{{\displaystyle 14}}}}\end{array}$

$\begin{array}{rl}x+y-z& =-1\\ -2+y-14& =-1\\ y-16& =-1\\ y& =\boxed{{\displaystyle \boxed{{\displaystyle 15}}}}\end{array}$

Jadi, himpunan penyelesaian dari SPLTV $$\{\begin{array}{ll}x+y-z& =-1\\ 2x-2y+3z& =8\\ 2x-y+2z& =9\end{array}$$ adalah (-2,15,14).
JAWAB: D