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Bab 2. Program LinearKelas 11, Agustus – September 2019
→ mempelajari cara mendapat nilai optimum (maks - min) dari
bbrp model pertidaksamaan (= sistem pertidaksamaan).
Yang harus dikuasai: Ø Pers garis lurus
Ø Pertidaksamaan linear 2 var
𝑨. 𝑷𝒆𝒓𝒔. 𝒈𝒂𝒓𝒊𝒔 𝒍𝒆𝒘𝒂𝒕 𝟐 𝒕𝒊𝒕𝒊𝒌 𝒃𝒆𝒃𝒂𝒔
𝑥1, 𝑦1
𝑥4, 𝑦4
∆𝒙
∆𝒚
𝑔𝑟𝑎𝑑𝑖𝑒𝑛:
𝑝𝑒𝑟𝑠 𝑔𝑎𝑟𝑖𝑠:
𝑎𝑡𝑎𝑢:
∆𝑦∆𝑥
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
𝑦 = 𝑚𝑥 + 𝒄
𝒄
1) 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑝𝑒𝑟𝑠 𝑔𝑎𝑟𝑖𝑠 𝑦𝑔 𝑚𝑒𝑙𝑎𝑙𝑢𝑖:
𝑎) 4, 2 & (1, 8)
S(4, 2)(1, 8) 𝑚 =
2 − 84 − 1 = −2
𝑦 − 8 = −2(𝑥 − 1)
𝑦 = −2𝑥 + 10
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 𝑦 = 𝑚𝑥 + 𝒄
2 = −2 . 4 + 𝒄10 = 𝒄
𝑦 = −2𝑥 + 10
𝑏) 1, 13 & (7, 5)
S(1, 13)(7, 5) 𝑚 =
13 − 51 − 7 = −
43
𝑦 − 5 = −43 (𝑥 − 7)
𝑘𝑎𝑙𝑖 3 → 3𝑦 − 15 = −4(𝑥 − 7)
3𝑦 + 4𝑥 = 43
𝑩. 𝑷𝒆𝒓𝒔. 𝒈𝒂𝒓𝒊𝒔 𝒍𝒆𝒘𝒂𝒕 𝒕𝒊𝒕𝒊𝒌 𝒅𝒊 𝒔𝒃. 𝒙 & 𝒔𝒃. 𝒚
𝒂
𝒃
𝒂𝑥 + 𝒃𝑦 = 𝒂𝒃
“ 𝑎𝑛𝑔𝑘𝑎 𝑑𝑖 𝑠𝑏. 𝑦 𝑘𝑎𝑙𝑖 𝑥𝑡𝑎𝑚𝑏𝑎ℎ
𝑎𝑛𝑔𝑘𝑎 𝑑𝑖 𝑠𝑏. 𝑥 𝑘𝑎𝑙𝑖 𝑦= ℎ𝑎𝑠𝑖𝑙 𝑘𝑎𝑙𝑖 𝑘𝑒𝑑𝑢𝑎𝑛𝑦𝑎 “
𝑟𝑢𝑚𝑢𝑠 𝑏𝑢𝑙𝑎𝑛:𝐽𝑜𝑛𝑜
𝑝𝑒𝑚𝑏𝑢𝑘𝑡𝑖𝑎𝑛 ?
(0, 𝒂)
(𝒃, 0)
𝒂𝑥 + 𝒃𝑦 = 𝒂𝒃
𝑚 =𝒂 − 00 − 𝒃 = −
𝒂𝒃
𝑦 − 𝒂 = −𝒂𝒃 𝑥 − 0
𝒃𝑦 − 𝒂𝒃 = −𝒂𝑥 →
𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑝𝑒𝑟𝑠 𝑔𝑎𝑟𝑖𝑠 𝐿1, 𝐿2, 𝐿3
𝐿1 ≡
𝐿2 ≡
5𝑥 + 7𝑦 = 35
4𝑥 − 2𝑦 = −8 → 2𝑥 − 𝑦 = −4
S(−1, 1)(4, 4) 𝑚 =
1 − 4−1 − 4
=35
𝑦 − 4 =35(𝑥 − 4) → 5𝑦 = 3𝑥 + 8
c 3
5
𝑪. 𝑴𝒆𝒏𝒈𝒈𝒂𝒎𝒃𝒂𝒓 𝒈𝒂𝒓𝒊𝒔
1) 𝐺𝑎𝑚𝑏𝑎𝑟𝑘𝑎𝑛 𝑦 = 2𝑥 + 4
𝒙 𝒚0
04
−2→ (0, 4)→ (−2, 0)
𝑦=2𝑥+4
2) 3𝑥 + 2𝑦 = 10
𝒙 𝒚0
05
10/3
𝑫. 𝑴𝒆𝒏𝒈𝒈𝒂𝒎𝒃𝒂𝒓 𝒅𝒂𝒆𝒓𝒂𝒉 𝒑𝒆𝒓𝒕𝒊𝒅𝒂𝒌𝒔𝒂𝒎𝒂𝒂𝒏
1) 𝐺𝑎𝑚𝑏𝑎𝑟𝑘𝑎𝑛 𝑦 ≤ 2𝑥 + 4
𝒙 𝒚0
04
−2
𝑐𝑒𝑘 𝑡𝑖𝑡𝑖𝑘 (0, 0)
0 ≤ 2 . 0 + 4?
𝐵𝐸𝑁𝐴𝑅𝐻𝑃
2) 𝐺𝑎𝑚𝑏𝑎𝑟𝑘𝑎𝑛 2𝑦 + 𝑥 < 0
𝒙 𝒚0 04 −2
𝑐𝑒𝑘 (1, 0)
2 . 0 + 1 < 0?
𝑆𝐴𝐿𝐴𝐻
×
×
×
×
×𝐻𝑃
3) 𝑥 + 2𝑦 ≥ 6 ; 3𝑥 + 2𝑦 < 12 ; & 𝑥 ≥ 0
𝒙 𝒚0 36 0
𝒙 𝒚0 64 0
×
××
××
×
×
𝑐𝑒𝑘 (0, 0)×
×
×
×
×
×
×
×
𝐻𝑃
×
×
×
×
×
×
4) 2𝑥 + 3𝑦 ≤ 12 ; 2𝑥 + 𝑦 ≥ 8 ; 𝑦 ≥ 0
xx
x
x
x
xx
x
x
x
x
x
x
x
xx x x x
HPx
𝐾𝑒𝑟𝑗𝑎𝑘𝑎𝑛 𝑈𝐾 2.2 ℎ𝑎𝑙𝑚 46
𝟑𝒂) 𝑥 + 𝑦 ≤ 6
2𝑥 + 𝑦 ≤ 10
5𝑥 + 9𝑦 ≤ 45
𝑥 ≥ 0 ; 𝑦 ≥ 0
𝐸.𝑀𝑒𝑛𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑝𝑒𝑟𝑡𝑖𝑑𝑎𝑘𝑠𝑎𝑚𝑎𝑎𝑛 𝑑𝑎𝑟𝑖 𝑔𝑎𝑚𝑏𝑎𝑟
1) 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑠𝑖𝑠𝑡𝑒𝑚:
6𝑥 + 2𝑦 = 12
3𝑥 + 𝑦 ≤ 6
𝑥 ≥ 0
𝑦 ≥ 0
2) 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑠𝑖𝑠𝑡𝑒𝑚:
𝑨
5𝑥 + 3𝑦 15≥
𝑦 ≥ 1
𝑨 →
5𝑥 + 6𝑦 30≤𝑩 →
𝑩
3.
–x + y ≥ –6
x + y ≥ –2
x ≥ 0y ≤ 0
−6𝑥 + 6𝑦 − 36≥𝑨 →
𝑨
−2𝑥 − 2𝑦 4≤𝑩 →
𝑩
4.3x + 2y ≥ 6
2x + 3y ≤ 6
-x + 2y ≥ -4
= fungsi dari tujuan yg dicari
Nilai optimum terletak di salah satu
titik pojok dari daerah yg diarsir
Optimum / Obyektif Fungsi F.
1) Tentukan nilai maks & min f(x, y) = 5x - 4y
𝟓𝒙 − 𝟒𝒚(2, 0)(3,2)(2, 3)(0, 2)(0, 0)
𝑚𝑎𝑘𝑠
𝑚𝑖𝑛
𝟓𝒙 − 𝟒𝒚(2, 0) 10(3,2) 7(2, 3) −2(0, 2) −8(0, 0) 0
2) maks & min: z = 2x + 3y
𝟐𝒙 + 𝟑𝐲(4, 0) 8(8,0) 16(2, 3) 13
𝑚𝑎𝑘𝑠
𝑚𝑖𝑛
ïïþ
ïïý
ü
³³£+£+
00462
yxyxyx3. Dari sistem:
tentukan nilai maks z = 3x + 2y
yxyxf
yxyxyx
23),(
00462
+=
ïïþ
ïïý
ü
³³£+£+
3x + 2y
(3, 0)
(2, 2)
(0, 4)
(0, 0)
9
10
8
0
ümaks
𝐾𝑒𝑟𝑗𝑎𝑘𝑎𝑛 𝑈𝐾 2.4 ℎ𝑎𝑙𝑚 51 − 52
1𝑎) 3)
𝑥 + 3𝑦 ≥ 4𝑦 − 𝑥 ≤ 2
𝑥 + 2𝑦 ≤ 10𝑥 ≥ 0 ; 𝑦 ≥ 0
𝑛𝑖𝑙𝑎𝑖 𝑜𝑝𝑡𝑖𝑚𝑢𝑚𝑧 = 3𝑥 + 4𝑦
𝑓(𝑥, 𝑦) = 𝑥 + 3𝑦
x + 3y4, 0 4
10, 0 102, 4 140, 2 6
0, 4/3 4
1𝑎) 𝑥 + 3𝑦 ≥ 4𝑦 − 𝑥 ≤ 2
𝑥 + 2𝑦 ≤ 10𝑥 ≥ 0 ; 𝑦 ≥ 0
𝑓(𝑥, 𝑦) = 𝑥 + 3𝑦
4/3
4
××
××
××
××
×
2
−2
×
××
××
××
×
5
10
××
×
××
×
××
××
××
××
××
𝐴
𝑥 + 2𝑦 = 10−𝑥 + 𝑦 = 2 � 𝐴(2, 4)
𝑚𝑎𝑥 = 14 ; 𝑚𝑖𝑛 = 4
3) 𝑛𝑖𝑙𝑎𝑖 𝑜𝑝𝑡𝑖𝑚𝑢𝑚 𝑧 = 3𝑥 + 4𝑦
7𝑥 + 6𝑦 = 429𝑥 + 5𝑦 = 45
𝐴6019
,6319�
3x + 4y3, 0 9
5, 0 15
60/19, 63/19 432/19
0, 7 28
0, 2 8
𝑚𝑎𝑥 = 28 ; 𝑚𝑖𝑛 = 8
𝐺. 𝑆𝑜𝑎𝑙 𝑐𝑒𝑟𝑖𝑡𝑎
1) 𝐽𝑒𝑛𝑖 𝑚𝑒𝑛𝑗𝑢𝑎𝑙 𝑘𝑢𝑒 𝐴 & 𝐵.
𝐾𝑢𝑒 𝐴 𝑑𝑖𝑏𝑒𝑙𝑖 𝑅𝑝 4.000, − & 𝑑𝑖𝑗𝑢𝑎𝑙 𝑅𝑝 7.500, −
𝐾𝑢𝑒 𝐵 𝑑𝑖𝑏𝑒𝑙𝑖 𝑅𝑝 2.000, − & 𝑑𝑖𝑗𝑢𝑎𝑙 𝑅𝑝 4.000, −
𝑀𝑜𝑑𝑎𝑙𝑛𝑦𝑎 ℎ𝑎𝑛𝑦𝑎 𝑅𝑝 400.000, − &
𝑖𝑎 𝑚𝑎𝑘𝑠𝑖𝑚𝑢𝑚 𝑚𝑒𝑛𝑗𝑢𝑎𝑙 140 𝑘𝑢𝑒.
𝐻𝑖𝑡𝑢𝑛𝑔 𝑘𝑒𝑢𝑛𝑡𝑢𝑛𝑔𝑎𝑛 𝑚𝑎𝑘𝑠 𝑛𝑦𝑎.
𝐴 𝐵𝐵𝑒𝑙𝑖𝐽𝑢𝑎𝑙
𝑈𝑛𝑡𝑢𝑛𝑔
40007500𝟑𝟓00
20004000𝟐𝟎00
400.000140
→→ 𝑥 + 𝑦 ≤ 140
2𝑥 + 𝑦 ≤ 200
𝑥 = 60𝑦 = 80
� (60, 80)
100, 0
60, 80
0, 140
𝟑𝟓𝑥 + 𝟐𝟎𝑦
3500
2100 + 1600
2800
𝑢𝑛𝑡𝑢𝑛𝑔 𝑚𝑎𝑥 = 𝑅𝑝 370.000,−
𝒛
2) 𝑈𝑛𝑡𝑢𝑘 𝑏𝑖𝑘𝑖𝑛 𝑡𝑎𝑠 𝑠𝑒ℎ𝑎𝑟𝑔𝑎 𝑅𝑝 200.000, −𝑝𝑒𝑟𝑙𝑢 𝑏𝑎ℎ𝑎𝑛 3 𝑘𝑔 & 𝑤𝑎𝑘𝑡𝑢 18 𝑗𝑎𝑚.
𝑈𝑛𝑡𝑢𝑘 𝑏𝑖𝑘𝑖𝑛 𝑠𝑒𝑝𝑎𝑡𝑢 𝑠𝑒ℎ𝑎𝑟𝑔𝑎 𝑅𝑝 300.000, −𝑝𝑒𝑟𝑙𝑢 𝑏𝑎ℎ𝑎𝑛 2 𝑘𝑔 & 𝑤𝑎𝑘𝑡𝑢 24 𝑗𝑎𝑚.
𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑚𝑎𝑘𝑠 𝑝𝑒𝑛𝑗𝑢𝑎𝑙𝑎𝑛 𝑗𝑖𝑘𝑎 𝑘𝑒𝑟𝑗𝑎720 𝑗𝑎𝑚 𝑗𝑖𝑘𝑎 𝑠𝑡𝑜𝑘 𝑏𝑎ℎ𝑎𝑛 90 𝑘𝑔.
𝑡𝑎𝑠 𝑠𝑒𝑝𝑎𝑡𝑢
𝑏𝑎ℎ𝑎𝑛
𝑤𝑎𝑘𝑡𝑢𝑗𝑢𝑎𝑙
3
18𝟐𝟎𝟎 𝑟𝑏
2
24𝟑𝟎𝟎 𝑟𝑏
90
720 →→ 3𝑥 + 2𝑦 ≤ 90
3𝑥 + 4𝑦 ≤ 120
𝑦 = 15𝑥 = 20
� (20, 15)
30, 0
20, 15
0, 30
𝟐𝟎𝟎𝑥 + 𝟑𝟎𝟎𝑦
6000
4000 + 4500
9000
𝑗𝑢𝑎𝑙 𝑚𝑎𝑥 = 𝑅𝑝 9 𝑗𝑢𝑡𝑎, −
𝒛
3) 𝐷𝑟. 𝐽𝑜 𝑚𝑒𝑛𝑦𝑎𝑟𝑎𝑛𝑘𝑎𝑛 𝐴𝑛𝑒 𝑢𝑛𝑡𝑢𝑘 𝑚𝑖𝑛𝑢𝑚
𝑚𝑖𝑛𝑖𝑚𝑎𝑙 10 𝑢𝑛𝑖𝑡 𝑣𝑖𝑡. 𝐴 & 8 𝑣𝑖𝑡. 𝐵 𝑑𝑎𝑙𝑎𝑚
𝑡𝑎𝑏𝑙𝑒𝑡 & 𝑘𝑎𝑝𝑠𝑢𝑙.
𝑇𝑖𝑎𝑝 𝑡𝑎𝑏𝑙𝑒𝑡 𝑏𝑒𝑟𝑖𝑠𝑖 2 𝑣𝑖𝑡. 𝐴 & 1 𝑣𝑖𝑡. 𝐵
𝐾𝑎𝑝𝑠𝑢𝑙 𝑏𝑒𝑟𝑖𝑠𝑖 1 𝑣𝑖𝑡. 𝐴 & 2 𝑣𝑖𝑡. 𝐵
𝐻𝑎𝑟𝑔𝑎 𝑡𝑎𝑏𝑙𝑒𝑡 𝑅𝑝 800/𝑏𝑢𝑎ℎ & 𝑘𝑎𝑝𝑠𝑢𝑙 600/𝑏𝑢𝑎ℎ
𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑏𝑎𝑛𝑦𝑎𝑘𝑛𝑦𝑎 𝑡𝑎𝑏𝑙𝑒𝑡 & 𝑘𝑎𝑝𝑠𝑢𝑙 𝑦𝑔
𝑚𝑒𝑠𝑡𝑖 𝑑𝑖𝑏𝑒𝑙𝑖 𝑎𝑔𝑎𝑟 𝑖𝑟𝑖𝑡 𝑏𝑖𝑎𝑦𝑎.
𝐴 𝐵
𝑡𝑎𝑏𝑙𝑒𝑡
𝑘𝑎𝑝𝑠𝑢𝑙
2
110
1
28
800
600
2𝑥 + 𝑦 ≥ 10𝑥 + 2𝑦 ≥ 8
(4, 2)
𝒛
8x + 6y8, 0 644, 2 44
0, 10 60
𝑖𝑎 ℎ𝑎𝑟𝑢𝑠 𝑏𝑒𝑙𝑖 4 𝑡𝑎𝑏𝑙𝑒𝑡 & 2 𝑘𝑎𝑝𝑠𝑢𝑙
𝑚𝑎𝑡𝑒𝑟𝑖 𝑠𝑒𝑙𝑒𝑠𝑎𝑖 . . . . . .
𝐾𝑒𝑟𝑗𝑎𝑘𝑎𝑛 𝑈𝐾 2.6 ℎ𝑎𝑙𝑚 60
2) 3)
𝑈𝐾 2.6 ℎ𝑎𝑙𝑚 602) 𝑆𝑢𝑎𝑡𝑢 𝑝𝑎𝑏𝑟𝑖𝑘 𝑏𝑖𝑘𝑖𝑛 2 𝑗𝑒𝑛𝑖𝑠 𝑚𝑖𝑛𝑢𝑚𝑎𝑛.𝑀𝑖𝑛𝑢𝑚𝑎𝑛 𝐴 𝑏𝑢𝑡𝑢ℎ 45 𝑚𝑒𝑛𝑖𝑡 𝑑𝑖 𝑚𝑒𝑠𝑖𝑛 𝐼 &
40 𝑚𝑒𝑛𝑖𝑡 𝑑𝑖 𝑚𝑒𝑠𝑖𝑛 𝐼𝐼
𝐾𝑒𝑢𝑛𝑡𝑢𝑛𝑔𝑎𝑛 𝑚𝑖𝑛𝑢𝑚𝑎𝑛 𝐴 𝑅𝑝 6.500, − &𝑚𝑖𝑛𝑢𝑚𝑎𝑛 𝐵 𝑅𝑝 8.000, −
𝐽𝑖𝑘𝑎 𝑎𝑑𝑎 40 𝑗𝑎𝑚 𝑘𝑒𝑟𝑗𝑎 𝑚𝑒𝑠𝑖𝑛 𝐼 & 3𝟒 𝑗𝑎𝑚𝑚𝑒𝑠𝑖𝑛 𝐼𝐼, 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛 𝑘𝑒𝑢𝑛𝑡𝑢𝑛𝑔𝑎𝑛 𝑚𝑎𝑘𝑠.
𝑀𝑖𝑛𝑢𝑚𝑎𝑛 𝐵 𝑏𝑢𝑡𝑢ℎ 60 𝑚𝑒𝑛𝑖𝑡 𝑑𝑖 𝑚𝑒𝑠𝑖𝑛 𝐼 &50 𝑚𝑒𝑛𝑖𝑡 𝑑𝑖 𝑚𝑒𝑠𝑖𝑛 𝐼𝐼
!" 2.6 ℎ'() 602) ,-'.- /'0123 02324 2 56427 )24-)'4.824-)'4 9 0-.-ℎ 45 )642. <2 )6724 = &
40 )642. <2 )6724 ==
"6-4.-4?'4 )24-)'4 9 @/ 6.500, − &)24-)'4 C @/ 8.000, −
E23' '<' 40 5') 3615' )6724 = & 3G 5'))6724 ==, .64.-3'4 36-4.-4?'4 )'37.
824-)'4 C 0-.-ℎ 60 )642. <2 )6724 = &50 )642. <2 )6724 ==
𝐴 𝐵
𝐼
𝐼𝐼
45
406500
60
508000
2400
2040𝒛
→→ 3𝑥 + 4𝑦 ≤ 160
4𝑥 + 5𝑦 ≤ 204
𝑥 = 16𝑦 = 28
� (16, 28)
6500x + 8000y51, 0 331.500
16, 28 328.0000, 40 320.000
𝑢𝑛𝑡𝑢𝑛𝑔 𝑚𝑎𝑘𝑠𝑅𝑝 331.500,−
𝑈𝐾 2.6 ℎ𝑎𝑙𝑚 603) 𝐴𝑑𝑎 2 𝑚𝑎𝑐𝑎𝑚 𝑚𝑎𝑘𝑎𝑛𝑎𝑛 𝑏𝑒𝑟𝑛𝑢𝑡𝑟𝑖𝑠𝑖.𝑇𝑖𝑎𝑝 𝑜𝑛𝑠 𝑚𝑎𝑘𝑎𝑛𝑎𝑛 𝐼 𝑚𝑒𝑛𝑔𝑎𝑛𝑑𝑢𝑛𝑔 2 𝑣𝑖𝑡. 𝐴
& 1 𝑣𝑖𝑡. 𝐵 𝑠𝑒ℎ𝑎𝑟𝑔𝑎 𝑅𝑝 5.500, −
𝐵𝑒𝑟𝑎𝑝𝑎 𝑜𝑛𝑠 𝑡𝑖𝑎𝑝 𝑗𝑒𝑛𝑖𝑠 𝑚𝑎𝑘𝑎𝑛𝑎𝑛 𝑦𝑔
𝑇𝑖𝑎𝑝 𝑜𝑛𝑠 𝑚𝑎𝑘𝑎𝑛𝑎𝑛 𝐼𝐼 𝑚𝑒𝑛𝑔𝑎𝑛𝑑𝑢𝑛𝑔 5 𝑣𝑖𝑡. 𝐴& 4 𝑣𝑖𝑡. 𝐵 𝑠𝑒ℎ𝑎𝑟𝑔𝑎 𝑅𝑝 8.000, −
𝑚𝑒𝑠𝑡𝑖 𝑑𝑖𝑏𝑒𝑙𝑖 𝑢𝑛𝑡𝑢𝑘 𝑚𝑒𝑚𝑒𝑛𝑢ℎ𝑖 𝑘𝑒𝑏𝑢𝑡𝑢ℎ𝑎𝑛𝑚𝑖𝑛𝑖𝑚𝑢𝑚 25 𝑣𝑖𝑡. 𝐴 & 15 𝑣𝑖𝑡. 𝐵
𝐴 𝐵
𝐼
𝐼𝐼
2
525
1
415
5.500
8.000𝒛
2𝑥 + 5𝑦 ≥ 25𝑥 + 4𝑦 ≥ 15
253,53
𝟓𝟓𝒙 + 𝟖𝟎𝒚
15, 0 82.500
0, 5 𝟒𝟎. 𝟎𝟎𝟎253,53
1775003
𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = 𝟒𝟎. 𝟎00,−
𝑏𝑒𝑙𝑖 0 𝑜𝑛𝑠 𝑚𝑎𝑘𝑎𝑛𝑎𝑛 𝐼& 5 𝑜𝑛𝑠 𝑚𝑎𝑘𝑎𝑛𝑎𝑛 𝐼𝐼