Chapter 3 System of Linear Equations. 3.1 Linear Equations in Two Variables Forms of Linear Equation...

Chapter 3Chapter 3System of Linear System of Linear

EquationsEquations

3.1 Linear Equations in 3.1 Linear Equations in Two VariablesTwo Variables

Forms of Linear EquationForms of Linear Equation ax + by = c (a, b, c are constants)ax + by = c (a, b, c are constants) y = mx + b (b = y-intercept,y = mx + b (b = y-intercept,

m = slope) m = slope)

Solution of Solution of System of Linear EquationsSystem of Linear Equations

Given:Given:a)a) x + y = 7x + y = 7

b)b) x - y = -1x - y = -1 What are the coordinateWhat are the coordinate

values such that bothvalues such that bothequations are satisfied?equations are satisfied?

Answer: (3, 4)Answer: (3, 4)

x + y = 7

x – y = -1

Solution: Substitution Solution: Substitution MethodMethod Given:Given:

y = -2x + 4y = -2x + 47x – 2y = 37x – 2y = 3

Solve the system of equationsSolve the system of equations7x – 2(-2x + 4) = 37x – 2(-2x + 4) = 37x + 4x – 8 = 37x + 4x – 8 = 311 x = 1111 x = 11x = 1x = 1

y = -2(1) + 4 = 2y = -2(1) + 4 = 2 SolutionSolution

(1, 2)(1, 2) Check solutionCheck solution

Solution: Substitution Solution: Substitution MethodMethod Given:Given:

5x + 2y = 15x + 2y = 1x – 3y = 7x – 3y = 7

Solve the system of equationsSolve the system of equationsx = 3y + 7x = 3y + 75(3y + 7) + 2y = 15(3y + 7) + 2y = 115y + 35 + 2y = 115y + 35 + 2y = 117y = -3417y = -34y = -2y = -2

x = 3(-2) + 7 = 1x = 3(-2) + 7 = 1 SolutionSolution

(1, -2)(1, -2) Check solutionCheck solution

Solution: Eliminating One Solution: Eliminating One VariableVariable

Given:Given:3x + 4y = -103x + 4y = -105x – 2y = 185x – 2y = 18

Solve the system of equationsSolve the system of equations 3x + 4y = -10 3x + 4y = -102(5x – 2y) = 2(18)2(5x – 2y) = 2(18)

3x + 4y = -10 3x + 4y = -10 10x – 4y = 36 10x – 4y = 36 13x = 26 13x = 26 x = 2 x = 2

3(2) + 4y = -10 3(2) + 4y = -10 4y = -16 4y = -16 y = -4 y = -4

SolutionSolution(2, -4)(2, -4)

Lines with No Solutions Lines with No Solutions

GivenGiven3x – 2y = 63x – 2y = 66x – 4y = 186x – 4y = 18

-2(3x – 2y) = -2(6) -6x + 4y = 12-2(3x – 2y) = -2(6) -6x + 4y = 12 6x – 4y = 18 6x – 4y = 18 0 = 6 0 = 6

Are there values of (x, y) for which 0 Are there values of (x, y) for which 0 = 6?= 6?

Lines with No Solutions Lines with No Solutions

GivenGiven3x – 2y = 6 y = (2/3)x - 33x – 2y = 6 y = (2/3)x - 36x – 4y = 18 y = (2/3)x – 6x – 4y = 18 y = (2/3)x – (9/2)(9/2)

Your TurnYour Turn

Solve by substitution Solve by substitution methodmethod

a)a) x – 3y = 8x – 3y = 8y = 2x - 9y = 2x - 9

Solve by elimination Solve by elimination methodmethod

b)b) 2x + 3y = 62x + 3y = 62x – 3y = 62x – 3y = 6

a) x - 3(2x - 9) = 8x – 6x + 27 = 8-5x = -19x = 19/5y = 2(19/5) – 9 = 38/5 – 9 = (38 – 45)/5 = -7/5Solution: (19/5, -7/5)

b) 2x + 3y = 62x – 3y = 6

4x = 12x = 32(3) + 3y = 63y = 0y = 0Solution: (3, 0)

Your TurnYour Turn

Solve by elimination Solve by elimination methodmethod

2x – 5y = 132x – 5y = 135x + 3y = 175x + 3y = 17

3(2x – 5y) = 3(13)5(5x + 3y) = 5(17)

6x – 15y = 3925x + 15y = 85--------------------32x = 124x = 4

6(4) – 15y = 3924 – 15y = 39-15y = 15y = -1

Solution: (4, -1)

3.3 Systems of Linear 3.3 Systems of Linear EquationsEquations

in Three Variables in Three Variables {(x, y, z) | ax + by + cz = d}{(x, y, z) | ax + by + cz = d}

(x, y, z)

y-axis

z-axis

x-axis

Graph of Graph of Linear Equation in 3 VariablesLinear Equation in 3 Variables

{(x, y, z) | 2x + 3y + z = 1}{(x, y, z) | 2x + 3y + z = 1}Set of all points (x, y, z) in a Set of all points (x, y, z) in a particular plane in 3-D.particular plane in 3-D.

z-axis

A(1/2, 0, 0)

y-axis

x-axis

C(0, 0, 1)

B(0, 1/3, 0)

X-Y PlaneX-Y Plane

x-axis

y-axis

z-axis

{(x, y, z) | z = 0}

X-Z PlaneX-Z Plane

x-axis

y-axis

z-axis

{(x, y, z) | y = 0}

Y-Z PlaneY-Z Plane

x-axis

y-axis

z-axis

{(x, y, z) | x = 0}

Systems of Linear Equations in Systems of Linear Equations in 3 Variables3 Variables

Given:Given: air fare + hotel + car for $210air fare + hotel + car for $210 hotel + car for $112hotel + car for $112 air fare + hotel for $180air fare + hotel for $180

What is the cost of air fare only, hotel What is the cost of air fare only, hotel only, or car only?only, or car only?

SolutionSolution

Let x = air fareLet x = air fare y = hotel y = hotel z = car z = car

x + y + z = 210x + y + z = 210y + z = 112y + z = 112x + y = 180x + y = 180

Solution:Solution:(x, y, z) = (98, 82, 30)(x, y, z) = (98, 82, 30)

Solving Equations in 3 Solving Equations in 3 Variables (1)Variables (1)

Given: 5x – 2y – 4z = 3 Given: 5x – 2y – 4z = 3 3x + 3y + 2z = -3 3x + 3y + 2z = -3 -2x + 5y + 3z = 3 -2x + 5y + 3z = 3

5x–3y–4z=3 5x – 2y – 4z = 35x–3y–4z=3 5x – 2y – 4z = 33x+3y+2z=-3 2(3x + 3y + 2z) = 2(-3)3x+3y+2z=-3 2(3x + 3y + 2z) = 2(-3)

5x – 2y – 4z = 35x – 2y – 4z = 3 6x + 6y + 4z = -6 6x + 6y + 4z = -6 11x + 4y = -3 11x + 4y = -3

3x+3y+2z=-3 3(3x + 3y + 2z) = 3(-3)3x+3y+2z=-3 3(3x + 3y + 2z) = 3(-3)-2x+5y+3z=3 -2(-2x + 5y + 3z) = -2(3)-2x+5y+3z=3 -2(-2x + 5y + 3z) = -2(3)


11x + 4y = -311x + 4y = -3

3x+3y+2z=-3 3(3x + 3y + 2z) = 3(-3)3x+3y+2z=-3 3(3x + 3y + 2z) = 3(-3)-2x+5y+3z=3 -2(-2x + 5y + 3z) = -2(3)-2x+5y+3z=3 -2(-2x + 5y + 3z) = -2(3)

9x + 9y + 6z = -99x + 9y + 6z = -9 4x - 10y - 6z = -6 4x - 10y - 6z = -6 13x - y = -15 13x - y = -15

11x+4y=-3 11x + 4y = -311x+4y=-3 11x + 4y = -313x- y=-15 4(13x – y) = 4(-15)13x- y=-15 4(13x – y) = 4(-15)

11x + 4y = -311x + 4y = -3 52x – 4y = -60 52x – 4y = -60 63x = -63 63x = -63


63x = -6363x = -63 x = -1x = -1

We had: 11x + 4y = -3We had: 11x + 4y = -3 11(-1) + 4y = -3 11(-1) + 4y = -3 4y = -3 + 11 = 8 4y = -3 + 11 = 8 y = 2y = 2

We had: 3x + 3y + 2z = -3We had: 3x + 3y + 2z = -3 3(-1) + 3(2) + 2z = -3 3(-1) + 3(2) + 2z = -3 -3 + 6 + 2z = -3 -3 + 6 + 2z = -3 2z = -6 2z = -6 z = -3z = -3

Solution: (-1, 2, -3) Check the Solution: (-1, 2, -3) Check the solutionsolution


Given: x + z = 8Given: x + z = 8 x + y + 2z = 17 x + y + 2z = 17 x + 2y + z = 16 x + 2y + z = 16

x+y+2z = 17 -2(x + y + 2z) = -2(17)x+y+2z = 17 -2(x + y + 2z) = -2(17)x+2y+z = 16 x + 2y + z) = 16x+2y+z = 16 x + 2y + z) = 16

-2x - 2y - 4z = -34 -2x - 2y - 4z = -34 x + 2y + z = 16 x + 2y + z = 16 -x - 3z = -18 -x - 3z = -18

x + z = 8x + z = 8 -x - 3z = 18 -x - 3z = 18 2z = 10 2z = 10 z = 5z = 5


z = 5z = 5

We had: x + z = 8We had: x + z = 8 x + (5) = 8 x + (5) = 8 x = 3x = 3

We had: x + y + 2z = 17We had: x + y + 2z = 17 (3) + y + 2(5) = 17 (3) + y + 2(5) = 17 3 + y + 10 = 17 3 + y + 10 = 17 y = 4 y = 4

Solution: (3, 4, 5)Solution: (3, 4, 5)

Check your solutionCheck your solution

Incosistent SystemsIncosistent SystemsGiven: 2x + 5y + z = 12Given: 2x + 5y + z = 12 x – 2y + 4z = -10 x – 2y + 4z = -10 -3x + 6y – 12z = 20 -3x + 6y – 12z = 20

x-2y+4z=10 3(x – 2y + 4z) = 3(-x-2y+4z=10 3(x – 2y + 4z) = 3(-10) -3x+6y-12z=20 -3x + 6y – 12z = 10) -3x+6y-12z=20 -3x + 6y – 12z = 2020

3x – 6y + 12z = -303x – 6y + 12z = -30 -3x + 6y – 12z = 20 -3x + 6y – 12z = 20 0 = - 0 = -10 ???10 ???

•No (x, y, z) to satisfy this condition.No (x, y, z) to satisfy this condition.

•No common point of intersectionNo common point of intersection

•System of equations is inconsistentSystem of equations is inconsistent

Solution to air-hotel-car Solution to air-hotel-car problems problems

Let x = air fareLet x = air fare y = hotel y = hotel z = car z = car

a) x + y + z = 210a) x + y + z = 210b) y + z = 112b) y + z = 112c) x + y = 180c) x + y = 180

• a) x + y + z = 210b) y + z = 112c) x + y = 180

• Use a) and b) to eliminate z x + y + z = 210-1( y + z) = -(112)

x + y + z = 210 - y - z = -112------------------------ d) x = 98

• a) x + y + z = 210b) y + z = 112c) x + y = 180

• Use b) and c) to eliminate y y + z = 112 -(x + y) = -(180) y + z = 112 -x – y = -180------------------------e) -x + z = -68

• a) x + y + z = 210b) y + z = 112c) x + y = 180

• Use d) and e) to find zd) x = 98e) –x + z = -68

-(98) + z = -68z = -68 + 98z = 30

• a) x + y + z = 210b) y + z = 112c) x + y = 180

• Use c) and d) to find yc) x + y = 180d) x = 98

98 + y = 180y = 180 – 98 = 82

Solution: (98, 82, 30) … (x: air, y: hotel, z: car)

Your TurnYour Turn

Solve the following systemSolve the following system x + y + 2z = 11 x + y + 2z = 11 x + y + 3z = 14 x + y + 3z = 14 x + 2y – z = 5 x + 2y – z = 5

Is it consistent or inconsistentIs it consistent or inconsistent

Your TurnYour Turn

a) x + y + 2z = 11a) x + y + 2z = 11b) x + y + 3z = 14b) x + y + 3z = 14c) x + 2y – z = 5c) x + 2y – z = 5

To rid of xTo rid of xa) –(x + y + 2z) = -(11) -x – y – 2z = -11a) –(x + y + 2z) = -(11) -x – y – 2z = -11b) x + y + 3z = 14 x + y + 3z = 14b) x + y + 3z = 14 x + y + 3z = 14 ----------------- ----------------- z = 3 z = 3a) x + y + 2(3) = 11 x + y = 5 a) x + y + 2(3) = 11 x + y = 5

b) x + y + 3(3) = 14 x + y = 9b) x + y + 3(3) = 14 x + y = 9

5 = 9 No values of x, y will make this true.5 = 9 No values of x, y will make this true. Thus, no solution. (Inconsistent system) Thus, no solution. (Inconsistent system)

3.4 Matrix Solution to 3.4 Matrix Solution to Linear SystemsLinear Systems

MatrixMatrix Arrangement of number in rows and Arrangement of number in rows and

columnscolumns Population (in mil.)Population (in mil.)

2000 2002 2004 2006 2000 2002 2004 2006men 52 54 55 57 men 52 54 55 57 women 53 54 54 57women 53 54 54 57

Augmented MatrixAugmented Matrix

4x + 4y = 194x + 4y = 19 2y + 3z = 8 2y + 3z = 84x - 5z = 74x - 5z = 7

4 4 0 19 1 a b c4 4 0 19 1 a b c 0 2 3 8 0 1 d e 0 2 3 8 0 1 d e 4 0 -5 7 0 0 1 f 4 0 -5 7 0 0 1 f

Matrix Row OperationsMatrix Row Operations

4x + 3y = -15 x + 2y = -1

4 3 -151 2 -1

1 a b0 1 c

•Interchange i th and j th rows: Ri Rj

•Multiply each element in the ith row by k: kRi

•Add k times the elements in row i to corresponding elements in Row j: kRi + Rj

Row-echelon form

Matrix Row OperationsMatrix Row Operations

4 3 -151 2 -1

3(4) 3(3) 3(-15)1 2 -1

Interchange R1 and R2: R1 R2

Multiply each element in the 1st row by 3: 3R1

Given a 2 x 3 matrix:

12 9 -451 2 -1

1 2 -14 3 -15

=

Matrix Row OperationsMatrix Row Operations4 3 -151 2 -1

4 3 -153(4)+1 3(3)+2 3(-15)-1

Add 3 times the elements in row 1 to corresponding elements in Row 2: 3R1 + R2

= 4 3 -1513 11 -46

Your TurnYour Turn3 -2 54 -3 1

Perform the following operation: R1 R2

4 -3 13 -2 5

Given:

Your TurnYour Turn3 -2 54 -3 1

Perform the following operation: 2R2

Given:

3 -2 52(4) 2(-3) 2(1)

3 -2 58 -6 2=

Your TurnYour Turn

Perform the following operation: 3R1 + R2

3 -2 53(3)+4 3(-2)-3 3(5)+1

=3 -2 513 -9 16

Given:3 -2 54 -3 1

Matrix Row OperationsMatrix Row Operations4x + 4y = 19 2y + 3z = 84x - 5z = 7

4 4 0 190 2 3 84 0 -5 7

R1 R2 0 2 3 84 4 0 194 0 -5 7

kR2 (k = -2)

0 2 3 8-8 -8 0 -38 4 -2 -8 -1

kR1+R3 (k = -1)

0 2 3 8-8 -8 0 -38 4 0 -5 7

Solving Linear System in 2 Solving Linear System in 2 VariablesVariables

4x – 3y = -15 x + 2y = -1

4 -3 -15 1 2 -1

4 3 -151 2 -1

1 a b0 1 c

Recall the strategy.

Then, the last row says: y = cThe first row says: x + ay = b

Solving thy Linear System in 2 Solving thy Linear System in 2 VariablesVariables

4x – 3y = -15 x + 2y = -1

4 -3 -15 1 2 -1

4 -3 -151 2 -1

1 2 -14 -3 -15

R1 R2 (to make e11 = 1)

1 2 -1 0 -11 -11

-4R1 + R2 (to make e21 = 0)

(-1/11)R2 (to make e12 = 1) 1 2 -1

0 1 1

Solving System in 2 Solving System in 2 VariablesVariables

y = 1

x + 2(1) = -1x = -3

Solution: (-3, 1)

1 2 -1 0 1 1

Check:4x – 3y = -154(-3) – 3(1) =-12 – 3 = -15

x + 2y = -1(-3) + 2(1) = -3 + 2 = -1

Your TurnYour Turn

Solve the following using matrix Solve the following using matrix method.method.

3x – 6y = 13x – 6y = 12x – 4y = 2/32x – 4y = 2/33 -6 1 2 -4 2/3

1 a b 0 1 c

(1/3)R1 (to make e11 = 1)(1/3)3 (1/3(-6) (1/3)1 2 -4 2/3

1 -2 1/3 2 -4 2/3

=

Your TurnYour Turn(1/3)3 (1/3(-6) (1/3)1 2 -4 2/3

1 -2 1/3 2 -4 2/3

=

2R1-R2 (to make e21 = 0)

1 -2 1/3 2(1)-2 2(-2)-4 2(1/3) – 2/3

1 -2 1/3 0 -8 0

=

(-1/8)R2 (to make e22 = 1)1 -2 1/3

(-1/8)0 (-1/8)(-8) (-1/8)0

1 -2 1/3 0 1 0

=

Your TurnYour Turn

1 -2 1/3 0 1 0

y = 0

x – 2y = 1/3x – 0 = 1/3x = 1/3

Solution: (1/3, 0)

Check:

3x – 6y = 13(1/3) – 6(0) = 11 = 1

2x – 4y = 2/32(1/3) – 4(0) = 2/32/3 – 0 = 2/32/3 = 2/3

Your TurnYour Turn

Solve the following using matrix Solve the following using matrix method.method.

-3x + 4y = 12-3x + 4y = 12 2x + y = 3 2x + y = 3

Solving System in 3 VariablesSolving System in 3 Variablesx + y + z = 210 y + z = 112x + y = 180

1 1 1 2100 1 1 1121 1 0 180

(-1)R1+R3 1 1 1 2100 1 1 1120 0 -1 -30

(-1)R3

x + y + z = 210 y + z = 112 z = 30

1 1 1 210 0 1 1 112 0 0 1 30

z = 30y + 30 = 112 → y = 82x + 82 + 30 = 210 → z = 98

Your TurnYour Turn

Solve the following system of Solve the following system of equationsequations 3x + y + 2z = 313x + y + 2z = 31

x + y + 2z = 19 x + y + 2z = 19 x + 3y + 2z = 25 x + 3y + 2z = 25

3.5: 3.5: Determinants 2 x 2 Matrix2 x 2 Matrix

a1 b1 a2 b2

Determinant Determinant a1 b1 a2 b2

= a1b2 – a2b1

DeterminantsDeterminants

2 4-3 -5

= 2(-5) – (-3)4 = -10 + 12 = 2

10 9 6 5

= ?

4 3-5 -8

= ?

Cramer’s RuleCramer’s RuleGiven:a1x + b1y = c1a2x + b2y = c2

c1 b1c2 b2

a1 b1a2 b2 a1 c1

a2 c2

a1 b1a2 b2

x =

y =

Dxx = ------- D

Dyy = ------- D

Cramer’s RuleCramer’s RuleGiven:5x - 4y = 26x - 5y = 1

2 -41 -5

5 -46 -5

5 26 1

5 -46 -5

x =

y =

-10 – (-4) -6= ------------ = ---- = 6 -25 – (-24) -1

5 – 12 -7= -------- = ---- = 7 -1 -1

(6, 7)

Your TurnYour Turn

Use Cramer’s rule to solveUse Cramer’s rule to solve1.1. 12x + 3y = 1512x + 3y = 15

2x – 3y = 132x – 3y = 13

2.2. x – 3y = 4x – 3y = 43x – 4y = 123x – 4y = 12

Solution for 1Solution for 1

12x + 3y = 1512x + 3y = 152x – 3y = 132x – 3y = 13

12 312 3D = 2 -3 = 12(-3) – 2(3) = -42D = 2 -3 = 12(-3) – 2(3) = -42

15 315 3DDxx = 12 -3 = 15(-3) – 3(12) = -84 = 12 -3 = 15(-3) – 3(12) = -84

DDxx -84 -84x = ---- = ----- = 2 x = ---- = ----- = 2 D -42 D -42


12x + 3y = 1512x + 3y = 152x – 3y = 132x – 3y = 13

12 312 3D = 2 -3 = 12(-3) – 2(3) = -42D = 2 -3 = 12(-3) – 2(3) = -42

12 1512 15DDyy = 2 13 = 12(13) – 2(15) = 126 = 2 13 = 12(13) – 2(15) = 126

DDyy 126 126y = ---- = ----- = -3y = ---- = ----- = -3 D -42 D -42

Solution: (2, -3)Solution: (2, -3)


x - 36 = 4x - 36 = 43x – 3y = 123x – 3y = 12

1 -31 -3D = 3 -3 = 1(-3) – 3(-3) = 6D = 3 -3 = 1(-3) – 3(-3) = 6

4 -34 -3DDxx = 12 -3 = 4(-3) – 12(-3) = 24 = 12 -3 = 4(-3) – 12(-3) = 24

DDxx 24 24y = ---- = ---- = 4y = ---- = ---- = 4 D 6 D 6


x - 36 = 4x - 36 = 43x – 3y = 123x – 3y = 12

1 -31 -3D = 3 -3 = 1(-3) – 3(-3) = 6D = 3 -3 = 1(-3) – 3(-3) = 6

1 41 4DDyy = 3 12 = 1(12) – 3(4) = 0 = 3 12 = 1(12) – 3(4) = 0

DDyy 0 0y = ---- = ---- = 0y = ---- = ---- = 0 D 6 D 6

Solution: (4, 0)Solution: (4, 0)

Determinant of a 3 x 3 Determinant of a 3 x 3 MatrixMatrix

aa1 b b1 c c1 b b2 c c2 b b1 c c1 b b1 c c1aa2 b b2 c c2 = a = a1 b b3 c c3 – a – a2 b b3 c c3 + a + a3 b b2 c c2aa3 b b3 c c3

E.g.E.g.

2 3 42 3 4 5 2 1 = ? 5 2 1 = ? 4 3 6 4 3 6

Determinant of a 3 x 3 Determinant of a 3 x 3 MatrixMatrix

Given: aGiven: a11x + bx + b11y + cy + c11z = dz = d11

a a22x + bx + b22y + cy + c22z = dz = d22

a a33x + bx + b33y + cy + c33z = dz = d33

aa11 b b11 c c11

D = aD = a22 b b22 c c22

a a33 b b33 c c33

dd11 b b22 c c33 D Dxx = d = d22 b b22 c c33

d d33 b b33 c c33

aa11 d d11 c c11 a a11 b b11 d d11

D Dyy = a = a22 d d22 c c2 2 DDzz = a = a22 b b22 d d22

a a33 d d33 c c3 3 aa3 3 bb3 3 dd33

DDxx

x = ----x = ---- D D

DDyy

y = ----y = ---- D D

D Dzz

z = ----z = ---- D D

ExampleExample

Given: x + 2y - z = -4Given: x + 2y - z = -4 x + 4y - 2z = -6 x + 4y - 2z = -6 2x + 3y + z = 3 2x + 3y + z = 3

1 2 -11 2 -1D = 1 4 -2D = 1 4 -2 2 3 1 2 3 1

-4 2 -1 -4 2 -1 D Dxx = -6 4 -2 = -6 4 -2 3 3 1 3 3 1

1 -4 -1 1 2 -41 -4 -1 1 2 -4 D Dyy = 1 -6 -2 = 1 -6 -2 DDzz = 1 4 -6 = 1 4 -6 2 3 1 2 3 1 2 3 32 3 3

ExampleExample 1 2 -1 4 -2 2 -1 2 -11 2 -1 4 -2 2 -1 2 -1D = 1 4 -2 = 1 3 1 - 1 3 1 + 2 4 -2D = 1 4 -2 = 1 3 1 - 1 3 1 + 2 4 -2 2 3 1 2 3 1

= (4 – (-6)) – (2 – (-3)) + 2(-4 – (-4)) = (4 – (-6)) – (2 – (-3)) + 2(-4 – (-4)) = 10 - 5 - 0 = 5 = 10 - 5 - 0 = 5

-4 2 -1 4 -2 2 -1 2 -1-4 2 -1 4 -2 2 -1 2 -1DDxx = -6 4 -2 = -4 3 1 - (-6) 3 1 + 3 4 -2 = -6 4 -2 = -4 3 1 - (-6) 3 1 + 3 4 -2 3 3 1 3 3 1

= (-4)(4 + 6) – (-6)(2 + 3) + (3)(-4 + 4)= (-4)(4 + 6) – (-6)(2 + 3) + (3)(-4 + 4) = -40 + 30 + 0 = 10 = -40 + 30 + 0 = 10

-10-10x = ----- = -2x = ----- = -2 5 5

Your TurnYour Turn

Calculate Calculate 1.1. y = Dy = Dyy / D / D

2.2. Z = DZ = Dz z / D/ D

Chapter 3 System of Linear Equations. 3.1 Linear Equations in Two Variables Forms of Linear Equation...

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Transcript of Chapter 3 System of Linear Equations. 3.1 Linear Equations in Two Variables Forms of Linear Equation...