SNBT Zone : Sistem Persamaan Aljabar
Table of Contents
Tipe:
No.
Jika A dan B memenuhi \begin{cases}\dfrac{3A}{2A+3B}+\dfrac{6B}{2A-3B}=3\\[8pt]\dfrac{-6A}{2A+3B}+\dfrac{3B}{2A-3B}=-1\end{cases} maka-\dfrac23 -\dfrac13 -\dfrac19
\dfrac19 \dfrac13
ALTERNATIF PENYELESAIAN
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{aligned}
6x+12y&=6\\
-6x+3y&=-1\qquad+\\\hline
15y&=5\\
y&=\dfrac13\\
\dfrac{B}{2A-3B}&=\dfrac13
\end{aligned}
\begin{aligned}
3x+6y&=3\\
3x+6\left(\dfrac13\right)&=3\\
3x+2&=3\\
x&=\dfrac13\\
\dfrac{A}{2A+3B}&=\dfrac13
\end{aligned}
\begin{aligned}
\dfrac{AB}{4A^2-9B^2}&=\left(\dfrac{A}{2A+3B}\right)\left(\dfrac{B}{2A-3B}\right)\\[4pt]
&=\left(\dfrac13\right)\left(\dfrac13\right)\\[4pt]
&=\dfrac19
\end{aligned}
Jadi, \dfrac{AB}{4A^2-9B^2}=\dfrac19 .
JAWAB: D
JAWAB: D
No.
Jika x dan y memenuhiALTERNATIF PENYELESAIAN
\begin{aligned}
\dfrac{2x+y}{3x-2y+3}&=\dfrac1{15}\\
15(2x+y)&=3x-2y+3\\
30x+15y&=3x-2y+3\\
27x+17y&=3
\end{aligned}
\begin{aligned}
\dfrac1{x+y}&=\dfrac7{-2x+y}\\
-2x+y&=7x+7y\\
-9x&=6y\\
3x+2y&=0
\end{aligned}
\begin{array}{rl|l}
27x+17y&=3&\times1\\
3x+2y&=0&\times9
\end{array}
\begin{aligned}
27x+17y&=3\\
27x+18y&=0\qquad-\\\hline
-y&=3\\
y&=-3
\end{aligned}
\begin{aligned}
3x+2y&=0\\
3x+2(-3)&=0\\
x&=2
\end{aligned}
\begin{aligned}
x-y&=2-(-3)\\
&=5
\end{aligned}
Jadi, x − y = 5.
No.
Jika- 28 − 32a
- 28 − 35a
- 28 − 38a
- 28 − 40a
- 28 − 46a
ALTERNATIF PENYELESAIAN
\begin{aligned}
x+by&=7&\qquad\color{red}{\times a}\\
ax+aby&=7a\\
ax+2y&=7a
\end{aligned}
\begin{aligned}
ax+y&=4&\qquad\color{red}{\times 7}\\
ax+2y&=7a&\qquad\color{red}{\times 5}
\end{aligned}
\begin{aligned}
7ax+7y&=28\\
5ax+10y&=35a&\qquad\color{red}{-}\\\hline
2ax-3y&=\boxed{\boxed{28-35a}}
\end{aligned}
Jadi, 2ax − 3y = 28 − 35a.
JAWAB: B
JAWAB: B
No.
Jika- −1
- 0
- 1
- 2
- 3
ALTERNATIF PENYELESAIAN
\begin{aligned}
\dfrac{216^x}{72^y}&= 162\\
\dfrac{\left(6^3\right)^x}{\left(2\cdot6^2\right)^y}&= 162\\
\dfrac{6^{3x}}{2^y\cdot6^{2y}}&= 162\\
\dfrac{6^{3x-2y}}{2^y}&= 2\cdot81\\
\dfrac{6^4}{2^y}&= 2\cdot3^4\\
2^y&=\dfrac{6^4}{2\cdot3^4}\\
&=\dfrac{2^4\cdot3^4}{2\cdot3^4}\\
&=2^3\\
y&=3
\end{aligned}
Jadi, nilai dari y adalah 3
JAWAB: E
JAWAB: E
No.
Jika a dan b memenuhi \begin{cases}\dfrac2{2a-b}+\dfrac7{2a+b}&=3\\\dfrac1{2a-b}-\dfrac7{2a+b}&=0\end{cases} maka- 5
- 6
- 7
- 8
- 9
ALTERNATIF PENYELESAIAN
\begin{aligned}
\dfrac2{2a-b}+\dfrac7{2a+b}&=3\\\dfrac1{2a-b}-\dfrac7{2a+b}&=0\qquad&{\color{red}+}\\\hline
\dfrac3{2a-b}&=3\\
2a-b&=1
\end{aligned}
\begin{aligned}
\dfrac2{2a-b}+\dfrac7{2a+b}&=3\\
\dfrac21+\dfrac7{2a+b}&=3\\
2+\dfrac7{2a+b}&=3\\
\dfrac7{2a+b}&=1\\
2a+b&=7
\end{aligned}
\begin{aligned}
2a-b&=1\\
2a+b&=7\qquad&{\color{red}+}\\\hline
4a&=8\\
2a&=4\\
a&=2
\end{aligned}
\begin{aligned}
2a+b&=7\\
4+b&=7\\
b&=3
\end{aligned}
\begin{aligned}
a+b&=2+3\\
&=\boxed{\boxed{5}}
\end{aligned}
Jadi, a + b = 5.
JAWAB: A
JAWAB: A
No.
Jumlah x dan y dari solusix − y = a
x2 + 5x − y = 2
adalah ....
- −12
- −10
- −6
- 6
- 10
ALTERNATIF PENYELESAIAN
\(\begin{aligned}
x-y&=a\\
y&=x-a
\end{aligned}\)
\(\begin{aligned} x^2+5x-y&=2\\ x^2+5x-(x-a)&=2\\ x^2+4x+a-2&=0 \end{aligned}\)
\(\begin{aligned} D&=4^2-4(1)(a-2)\\ &=16-4a+8\\ &=-4a+24 \end{aligned}\)
Kita lihat bahwa jikaD > 0 , ada tak hingga nilai a yang memenuhi, akibatnya ada tak hingga nilai x yang memenuhi. Kita sepakati bahwa sistem persamaan di atas mempunyai satu solusi sehingga D = 0.
\(\begin{aligned} x^2+5x-y&=2\\ x^2+5x-(x-a)&=2\\ x^2+4x+a-2&=0 \end{aligned}\)
\(\begin{aligned} D&=4^2-4(1)(a-2)\\ &=16-4a+8\\ &=-4a+24 \end{aligned}\)
Kita lihat bahwa jika
\(\begin{aligned}
-4a+24&=0\\
a&=6
\end{aligned}\)
\(\begin{aligned} x^2+4x+6-2&=0\\ x^2+4x+4&=0\\ (x+2)^2&=0\\ x&=-2 \end{aligned}\)
\(\begin{aligned} y&=x-a\\ &=-2-6\\ &=-8 \end{aligned}\)
\(\begin{aligned} x+y&=-2+(-8)\\ &=\boxed{\boxed{-10}} \end{aligned}\)
\(\begin{aligned} x^2+4x+6-2&=0\\ x^2+4x+4&=0\\ (x+2)^2&=0\\ x&=-2 \end{aligned}\)
\(\begin{aligned} y&=x-a\\ &=-2-6\\ &=-8 \end{aligned}\)
\(\begin{aligned} x+y&=-2+(-8)\\ &=\boxed{\boxed{-10}} \end{aligned}\)
Jadi, jumlah x dan y dari solusi (x, y) yang memenuhi sistem persamaan
x − y = a
x2 + 5x − y = 2
adalah −10.
JAWAB: B
x − y = a
x2 + 5x − y = 2
adalah −10.
JAWAB: B
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