SBMPTN Zone : Sistem Persamaan

Berikut ini adalah kumpulan soal mengenai Sistem Persamaan tingkat SBMPTN. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

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No. 1

Jika $A$ dan $B$ memenuhi $$\{\begin{array}{l}{\displaystyle \frac{3A}{2A+3B}}+{\displaystyle \frac{6B}{2A-3B}}=3\\ {\displaystyle \frac{-6A}{2A+3B}}+{\displaystyle \frac{3B}{2A-3B}}=-1\end{array}$$ maka $\dfrac{AB}{4A^2-9B^2}=$ ....

1. $>-\dfrac23$
2. $-\dfrac13$
3. $-\dfrac19$

4. $\dfrac19$
5. $\dfrac13$

SBMPTN 2017

Misal ${x=\dfrac{A}{2A+3B}}$ dan ${y=\dfrac{B}{2A-3B}}$

$\begin{array}{rl|l}3x+6y&=3&\times2\\-6x+3y&=-1&\times1\end{array}$

$\begin{array}{rl}6x+12y& =6\\ -6x+3y& =-1+\\ 15y& =5\\ y& ={\displaystyle \frac{1}{3}}\\ {\displaystyle \frac{B}{2A-3B}}& ={\displaystyle \frac{1}{3}}\end{array}$

$\begin{array}{rl}3x+6y& =3\\ 3x+6\left({\displaystyle \frac{1}{3}}\right)& =3\\ 3x+2& =3\\ x& ={\displaystyle \frac{1}{3}}\\ {\displaystyle \frac{A}{2A+3B}}& ={\displaystyle \frac{1}{3}}\end{array}$

$\begin{array}{rl}{\displaystyle \frac{AB}{4A^2-9B^2}}& =\left({\displaystyle \frac{A}{2A+3B}}\right)\left({\displaystyle \frac{B}{2A-3B}}\right)\\ & =\left({\displaystyle \frac{1}{3}}\right)\left({\displaystyle \frac{1}{3}}\right)\\ & ={\displaystyle \frac{1}{9}}\end{array}$

JAWAB: D

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No. 2

Jika $x$ dan $y$ memenuhi ${\dfrac{2x+y}{3x-2y+3}=\dfrac1{15}}$ dan ${\dfrac1{x+y}=\dfrac7{-2x+y}}$ maka nilai ${x-y=}$ ....

$\begin{array}{rl}{\displaystyle \frac{2x+y}{3x-2y+3}}& ={\displaystyle \frac{1}{15}}\\ 15(2x+y)& =3x-2y+3\\ 30x+15y& =3x-2y+3\\ 27x+17y& =3\end{array}$

$\begin{array}{rl}{\displaystyle \frac{1}{x+y}}& ={\displaystyle \frac{7}{-2x+y}}\\ -2x+y& =7x+7y\\ -9x& =6y\\ 3x+2y& =0\end{array}$

$\begin{array}{rl|l}27x+17y&=3&\times1\\3x+2y&=0&\times9\end{array}$

$\begin{array}{rl}27x+17y& =3\\ 27x+18y& =0-\\ -y& =3\\ y& =-3\end{array}$

$\begin{array}{rl}3x+2y& =0\\ 3x+2(-3)& =0\\ x& =2\end{array}$

$\begin{array}{rl}x-y& =2-(-3)\\ & =5\end{array}$

Jadi, $x-y=5$.

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No. 3

Jika ${ax+y=4}$, ${x+by=7}$, dan ${ab=2}$, maka nilai ${2ax-3y=}$

1. ${28-32a}$
2. ${28-35a}$
   
3. ${28-38a}$
   

4. ${28-40a}$
5. ${28-46a}$
   

$\begin{aligned}x+by&=7&\qquad\color{red}{\times a}\\ax+aby&=7a\\ax+2y&=7a\end{aligned}$

($$\begin{array}{rlr}ax+y& =4& \times 7\\ ax+2y& =7a& \times 5\end{array}$$\)

$\begin{aligned}7ax+7y&=28\\5ax+10y&=35a&\qquad\color{red}{-}\\\hline2ax-3y&=\boxed{\boxed{28-35a}}\end{aligned}$

Jadi, $2ax-3y=28-35a$.
JAWAB: B

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