5. Definisi
Sistem persamaan linear adalah kumpulan dua atau lebih
persamaan linear yang terdiri dari variabel-variabel yang
sama. Persamaan linear adalah persamaan matematika
yang dapat dinyatakan dalam bentuk ax + by + cz + ... = d,
di mana a, b, c, dan seterusnya adalah konstanta yang
dikenal sebagai koefisien, dan x, y, z, dan seterusnya
adalah variabel.
6. Bentuk
umum SPL
Atau dalam bentuk Ax = b
A = koefisien
X = matriks perubah
B = matriks konstanta
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
...
...
...
n n
n n
m m mn n m
a x a x a x b
a x a x a x b
a x a x a x b
8. Penyajian SPL dalam Matriks
2𝑥 − 5𝑦 + 𝑧
𝑥 + 3𝑦 − 𝑧
−2𝑦 − 2𝑧
2 −5 0
0 3 0
0 −2 −2
Jadi cara penyanjian sebuah sistem
persamaan linear ke dalam bentuk
matriks itu mengikuti X, Y , dan Z dari
sebuah sistem persamaan linear
9. AUGMENTED MATRIKS
Matriks yang diperluas (Augmented matrix) adalah matriks yang memiliki
ukuran 𝑚×(n+1), dengan m merupakan jumlah persamaan dalam SPL dan
(n+1) merupakan jumlah variabel yang ada pada SPL, ditambah dengan
sebuah kolom yang berisi konstanta yang bukan pengali variabel.
Sistem m persamaan linier dengan n buah variabel dapat diubah dalam
bentuk matriks yang diperluas (augmented matrix) yang dituliskan sebagai
berikut:
10. AUGMENTED MATRIKS
Diberikan Sistem Persamaan linear sebagai berikut :
𝑥 −𝑦+2𝑧=6
2𝑥+3𝑦 −𝑧=−3
−2𝑦 −3𝑧=−4
SPL tersebut dapat dituliskan dalam bentuk perkalian matriks, sebagai berikut:
1 −1 2
2 3 −1
0 −2 −3
𝑥
𝑦
𝑧
=
6
−3
−4
Dari langkah langkah diatas maka diperoleh matriks yang diperluasnya sebagai berikut :
𝐴𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑 =
1 −1 2
2 3 −1
0 −2 −3
6
−3
−4
14. SPL KONSISTEN
SPL Konsisten adalah SPL yang mempunyai penyelesaian. SPL Konsisten dapat dibedakan menjadi
2, yaitu : 1) SPL Independen (SPL dengan penyelesaian yang tunggal). Det 𝐴 ≠ 0
Contoh :
𝑥 − 𝑦 + 2𝑧 = 1
𝑥 + 𝑦 − 2𝑧 = 2
−𝑥 − 3𝑦 + 𝑧 = −2
Perkalian Matriksnya akan menjadi :
Maka penyelesaiannya adalah matriks konsisten solusi tunggal.
15. SPL KONSISTEN
SPL Dependen (SPL dengan solusi tak terhingga banyaknya). Det 𝐴 = 0 , 𝑘 = 0, 𝑙 = 0
Catatan :
19. MAIN VALUES
MERCURY
Mercury is the closest planet
to the Sun and the smallest
one in the Solar System
VENUS
Venus has a beautiful name
and is the second planet
from the Sun
20. WHOA!
This can be the part of the presentation where you
introduce yourself, write your email…
21. COEFFICIENT
MERCURY
It’s the closest planet to
the Sun and the smallest
in the Solar System
VENUS
Venus has a beautiful
name and is the second
planet from the Sun
MARS
Mars is actually a cold
place. It’s full of iron
oxide dust
22. ROOTS OF AN EQUATION
MERCURY
It’s the closest
planet to the Sun
VENUS
Venus it’s hot and
has a nice name
JUPITER
Jupiter is the
biggest planet
SATURN
It’s composed
mostly of hydrogen
27. 9h 55m 23s
Is Jupiter's rotation period
333,000
Earths is the Sun’s mass
386,000 km
Is the distance between Earth and the Moon
28. 13%
Jupiter is the biggest
planet of them all
25%
Venus is the second
planet from the Sun
62%
Mars is actually a
very cold place
PERCENTAGES
29. TABLET
APP
You can replace the image on
the screen with your own
work. Just right-click on it
and select “Replace image”
30. PHONE
WEB
You can replace the image
on the screen with your own
work. Just right-click on it
and select “Replace image”
31. LEARNING EXPONENTS
TAKE A EQUATION
LIKE THIS:
AND TURN IT
INTO THIS:
ax2 + bx + c = 0 a(x+d)2 + e = 0
TAKE A EQUATION
LIKE THIS:
AND TURN IT
INTO THIS:
x2 + ax + bx + ab = 0 x2+bx/a+(b/2a)2=-c/a+(b/2a)
32. MEET OUR CENTERS
MERCURY
It’s the closest
planet to the Sun
VENUS
Venus is the second
planet from the Sun
MARS
Mars is actually a
very cold place
33. SOLVING PROBLEMS
MERCURY
It’s the closest
planet to the Sun
VENUS
Venus is the second
planet from the Sun
MARS
Mars is actually a
very cold place
NEPTUNE
It’s the farthest
planet from the Sun
01 02 03 04
34. OVERVIEW DIAGRAM
JUPITER
Jupiter is the
biggest planet of
them all
Mars is actually a
very cold place
Mercury is the
smallest planet
Venus is the second
planet from the Sun
01
02
03
35. EFFECTIVENESS OF THIS METHOD
MERCURY
It’s the closest
planet to the Sun
MARS
Mars is actually a
very cold place
Follow the link in the graph to modify its data and then paste the new one here.
For more info, click here
36. FILL THE TABLE
GENERAL EQUATION FACTORIZATION
x2 + (a + b)x + ab = 0 x2 + 5x + 6 = 0
x2 + ax + bx + ab = 0 x2 + 2x + 3x + 6 = 0
x(x + a) + b(x + a) x(x + 2) + 3(x + 2) = 0
(x + a)(x + b) = 0 (x + 2)(x + 3) = 0
39. VALUES OF “D” AND “E”
Start with Ax2 + bx + c = 0
Divide the equation by a x2 = b/a x + c/a = 0
Put c/a on other side x2 = b/a x = - c/a
Add (b/2a)2 x2 = b/a x+ (b/2a)2=-c/a+(b/2a)2
Complete the square (x+b/2a)2=-c/a+(b/2a)2
To the left side (x+b/2a)2+c/a-(b/2a)2= 0
The original multiple a of x2 a(x+b/2a)2+c-b2/4a= 0
40. MIXED DEGREE SYSTEMS
WHAT ARE THE SOLUTIONS TO
THE FOLLOWING SYSTEMS?
{-2x² + y = -5
{ y = -3x² + 5
{ 10x² - y = 48
{ 2y = 16x² + 48
{ x + y = 11
{ 4x² - 3y² = 8
{ y = 6x² + 1
{ y = x² + 4
{ y = -2x²
(√2, -1)
42. 2
4
6
2 4 6 8 10 12
-2
-1
0
ONE TO ONE FUNCTION
𝑥 𝑓(𝑥)=√𝑥
0 𝑓(𝑥)=√0=0
1 𝑓(𝑥)=√1=1
2 𝑓(𝑥)=√2≈1.414
4 𝑓(𝑥)=√4=2
7 𝑓(𝑥)=√7≈2.645
9 𝑓(𝑥)=√9=3
16 𝑓(𝑥)=√16=4 y=√𝑥
43. ASSIGNMENTS
● This is a list of homework to learn the topic
● This is a list of homework to learn the topic
● This is a list of homework to learn the topic
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● This is a list of homework to learn the topic
● This is a list of homework to learn the topic
● This is a list of homework to learn the topic
● This is a list of homework to learn the topic
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