System of Linear Equation (SLE) Sub Chapter Introduction Solution of SLE using ERO Solution of SLE using Inverse matrix and Cramer’s Rule Homogeneous SLE Some Applications Electrical system Computer Network Economy Model etc. 28/11/2018 9:36

The problem can be expressed x + 2y = 5000 3x + y = 10000 Introduction Linear Equation is mathematical expression where the variable does not contain exponential, trygonometri (as sin, cos, dll.), or multiplication with another variable or itself. Example : If Ady buy 1 Laptop (x) and 2 PC (y) then he have to pay $ 5000, if he buy 3 Laptop and 1 PC then he have to buy $ 10000. The problem can be expressed x + 2y = 5000 3x + y = 10000 28/11/2018 9:36

Generally for of system of linear equation In the multiplication of matrix : 28/11/2018 9:36

Let SLE Or AX = B where Example : A is said coeffisien matrix X is said variable matrix B is said constant matrix Example : Let SLE x + 2y = 5000 3x + y = 10000 can be expressed in the form 28/11/2018 9:36

{x = 3000, y =1000 } is a solution of SLE We have SLE : x + 2y = 5000 3x + y = 10000 Then {x = 3000, y =1000 } is a solution of SLE {x = 1000, y =3000 } is not a solution of SLE A SLE has three possibility about solution : SLE has only one solution SLE has infinitely many solution SLE has no solution (inconstent) 28/11/2018 9:36

Ilustration of Solution in Cartesius SLE 2x – y = 2 x – y = 0 has one solution, i.e : x = 2, y = 2 x y y = 2x - 2 (2, 2) is crossing point of each line y = x 2 (2, 2) 1 2 28/11/2018 9:36

It can be expressed in cartesius SLE x – y = 0 2x – 2y = 2 It can be expressed in cartesius There is no crossing point It is mean that SLE has no solution y y = x y = x – 1 x 1 28/11/2018 9:36

SLE x – y = 0 2x – 2y = 0 2nd equation is twice of 1st equation It can be expressed in cartesius So, SLE has infinitely many solution y x – y = 0 2x – 2y = 0 x 28/11/2018 9:36

Get a solution of SLE using ERO SLE can be expressed in the form augmented matrix Using ERO to be reduced echelon row Example : Find a solution of SLE 3x – y = 5 x + 3y = 5 Answer : Augmented Matrix of SLE 28/11/2018 9:36

Solution of SLE is x = 2 and y = 1 Example : a. a + c = 4 a – b = –1 SLE expresion is Solution of SLE is x = 2 and y = 1 Example : Determine a solution (if possible) for SLE : a. a + c = 4 a – b = –1 2b + c = 7 28/11/2018 9:36

b. a + c = 4  a – b = –1 –a + b = 1 c. a + c = 4 –a + b = 2 Jawab : Solution of SLE is a = 1, b = 2, and c =3 28/11/2018 9:36

 b. or We have a + c = 4 dan b + c = 5. Let c = t, where t is a parameter. Solution of SLE is : , where t is a parameter 28/11/2018 9:36

We can see on 3rd row c.  we have 0.a + 0.b + 0.c = 1. It is not possible! Then there is no a, b, and c So, SLE has no solution. 28/11/2018 9:36

EXERCISES Solve system of linear equations below 28/11/2018 9:36

Determine a such that SLE : a. Has one solution b. Has no solution x + 2y – 3z = 4 3x – y + 5z = 2 4x + y + (a2 – 14) z = a+2 Determine a such that SLE : a. Has one solution b. Has no solution c. Has infinitely many solution 28/11/2018 9:36

Augmented Matrix of SLE Answer : Augmented Matrix of SLE a. SLE has one solution if: a2 – 16  0 sehingga a   4 28/11/2018 9:36

SLE has no solution when a2 – 16 = 0 dan a– 4  0 b. Focus on 3rd row 0x + 0y + (a2 – 16a) z = a – 4 SLE has no solution when a2 – 16 = 0 dan a– 4  0 such a =  4 and a  4. so , a = – 4. c. SLE has infinitely any solution a2 – 16 = 0 and a – 4 = 0 So , a = 4 28/11/2018 9:36

Get Solution SLE using inverse Matrix or AX = B If A is invertible then A–1 A X = A–1 B We have : X = A–1 B Remember that A is invertible if and only if Det (A)  0. 28/11/2018 9:36

Determine solution of SLE : a + c = 4 a – b = –1 2b + c = 7 Answer : Example : Determine solution of SLE : a + c = 4 a – b = –1 2b + c = 7 Answer : We know So, we have Inverse matrix 28/11/2018 9:36

X = A–1 B : so, Solution of SLE 28/11/2018 9:36

HOMOGENEOUS LINEAR EQUATIONS SYSTEM A system of linear equations is said to be homogeneous if the constant term are all zero; that is, the system has the form Every homogeneous system is consistent All system has x1=0,x2=0,…,xn=0 as a solution  trivial solution Denoted by If there are other solutions, they are called nontrivial solutions 28/11/2018 9:36

HOMOGENEOUS LINEAR EQUATIONS SYSTEM There are two only possibilities for homogeneous linear system’s solutions The system has only the trivial solution The system has infinitely many solutions in addition to the trivial solution Example Solve the homogeneous system of linear equations Augmented matrix Reduced row-echelon 28/11/2018 9:36

HOMOGENEOUS LINEAR EQUATIONS SYSTEM The solutions is infinitely many solutions / nontrivial solutions 28/11/2018 9:36

EXERCISES 2p + q – 2r – 2s = 0 p – q + 2r – s = 0 –p + 2q – 4r + s = 0 Find solutions of homogeneous linear system Ax = 0 where Find solutions of homogeneous linear system below 2p + q – 2r – 2s = 0 p – q + 2r – s = 0 –p + 2q – 4r + s = 0 3p – 3s = 0 28/11/2018 9:36

Get solution of SLE using Cramer’s Rule Let SLE is : The following sequence step for Cramer’s Rule : Find determinant of A (If det(A)=0, we cancel it) Determine Ai  matrix from A where ith column is replaced by B. Example: 28/11/2018 9:36

Find b of SLE : Calculate |Ai| Solution of SLE for variable xi is Example : Find b of SLE : a + c = 4 a – b = –1 2b + c = 7 Answer : 28/11/2018 9:36

So Solution of b is 2 28/11/2018 9:36

Find a solution of a ? Find a solution of c ? 28/11/2018 9:36

Example : Look the ilustration :  a b   c Show that : a2 = b2 + c2 – 2bc cos 28/11/2018 9:36

Answer : we have : c cos + b cos = a c cos + a cos = b b cos + a cos = c or 28/11/2018 9:36

Using Cramer’s rule : 28/11/2018 9:36

So, we have : a2 = b2 + c2 – 2bc cos 28/11/2018 9:36

1. Determine solution of SLE : Exercise 1. Determine solution of SLE : 2. Determine solution of SLE : 2p – 2q – r + 3s = 4 p – q + 2s = 1 –2p +2q – 4s = –2 3. Determine solution of Homogeneous SLE : 28/11/2018 9:36

5. Let 4. Let SLE AX = B Determine Solution of SLE using: ERO Inverse matrix Cramer’s rule 5. Let Find 28/11/2018 9:36

6. Let HSLE (with variable p, q, and r) Determine k such that SLE has one solution 7. Let Determine a not zero vector such that 28/11/2018 9:36