Section 8.1 System of Linear Equations: Substitution and ...

Section 8.1 System of Linear Equations: Substitution and Elimination

Identifying Linear Systems Linear Systems in two variables

Solution ,

0 0 Consistent/ Independent

Exactly one solution

Two lines cross at one point

1

Consistent/ Dependent Infinitely many solutions Two lines are identical

Linear Systems in Three Variables

Solution ,,

1

Inconsistent No solution Two lines are parallel

00

0

0

00

Consistent/ Independent

Exactly one solution

Three lines cross at one point

10 01

Consistent/ Dependent Infinitely many solutions Three lines are identical

10 01

Inconsistent No solution Three lines are parallel

Cheon-Sig Lee

coastalbend.edu/lee

Page 1

Section 8.1 System of Linear Equations: Substitution and Elimination

Solving Linear Systems by Substitution

Step 1: Solve either of the equations for one variable in terms of the other variable. Step 2: Substitute the expression obtained in step 1 into the other equation, then solve the

resulting equation containing one variable Step 3: Back-substitute the value obtained in step 3 into one of the original equations.

Example 1 Solve by the substitution method: 5

4 2

9 3

(Solution 1)

Step 1: Solve either of the equations for one variable in terms of the other variable.

From the second equation,

2

3

2

2

23

Step 2: Substitute the expression obtained in step 1 into the other equation, and then solve

resulting equation containing one variable.

549

23

52 3 4 9

10 15 4 9

6 15 9

15 15

6 24

6 24

66

4

Step 3: Back-substitute the value obtained in step 3 into one of the original equations.

Using the first equation

or

Using the second equation

549 4

2

3 4

5 44 9

24 3

5 16 9

83

16 16

88

5 25

5

5 25

5 5

5

Therefore, the solution is 5, 4 and two lines intersect at 5, 4 .

Cheon-Sig Lee

coastalbend.edu/lee

Page 2

Section 8.1 System of Linear Equations: Substitution and Elimination

Solving Linear Systems by Elimination

Step 1: Rewrite both equations in the form of Ax + By = C. Step 2: If necessary, multiply either equation or both equations by appropriate nonzero

numbers so that the x-coefficients or the y-coefficients are same. Step 3: Add or Substitute the equations obtained in step 2, then solve the resulting

equation in one variable. Step 4: Back-substitute the value obtained in step4 into one of the original equations.

Example 2: Solve by the elimination method: 5

4 2

9 3

(Solution 2)

Step 1: Rewrite both equations in the form of Ax + By = C. (Skip)

Step 2: If necessary, multiply either equation or both equations by appropriate nonzero

numbers so that the x-coefficients or the y-coefficients are same.

Multiply the second equation by 5. So, we have

54 9

5 10

15 5 5 2

35

Step 3: Add or Substitute the equations obtained in step 2, then solve the resulting equation in

one variable

549

5 10

15

6 24 6 24

66 4

Step 4: Back-substitute the value obtained in step 3 into one of the original equations

Using the first equation

or

549 4

5 44 9

Using the second equation

2

3 4

24 3

5 16 9

83

16 16 5 25 5 25

88 5

5 5 5

Therefore, the solution is 5, 4 and two lines intersect at 5, 4 .

Cheon-Sig Lee

coastalbend.edu/lee

Page 3

Section 8.1 System of Linear Equations: Substitution and Elimination

Solving Linear Systems by Graphing (TI-83)

5 4 9

Step1: Rewrite both equations in the form of y = mx + b.

2

3

Step2: Type the resulted equations in your calculator.

59 44 13 22

Step3: Graph and then find intercept. TI-83/84: graph and then 2nd TRACE .

TI-89: graph and then F5

Solving Linear Systems by Matrix Method (TI-83)

Step1: Rewrite both equations in the augmented matrix form

54 2

9 3

5 1

49 23

Step2: Type the matrix obtained in step1 in your calculator

TI-83/84: 2nd

x?1

> Edit > Dimension 2?3 >

5 1

4 2

9 3

> 2nd

MODE

> 2nd x?1 > MATH > rref > 2nd x?1 > [A]

TI-89: 2nd 5 > Matrix > rref ([5,4,9;1,-2,-3])

TI-83: Matrix Method

2nd

x?1

> Edit > Dimension 2?3 >

5 1

4 2

9 3

>

2nd

MODE > 2nd

x?1 > MATH > rref > 2nd

x?1 > [A]

Cheon-Sig Lee

coastalbend.edu/lee

Page 4

Section 8.1 System of Linear Equations: Substitution and Elimination

Exercises

1.

(Solution 1) Elimination Method Step 1:

Step 2: 2

Step 3: 5

23 322

2

23 422

55

55 10 5

15 15 15

14

Step 4: Substitute 2 for x in the equation or , then

6

solve for the variable z. From the equation ,

8

2

8

2

22

8

14

4

8

4

4

4

10

Step 5: Substitute 2 for x and 4 for z into one of the

10

original equations. From the equation ,

40

2

2

2, 4

30

22

42

30

15

Therefore, the solution is 2, 2, 4

Matrix Method 1 2 3 14

Augmented Matrix: 2 1 1 2 32 36

x

y z

Augmented Matrix Cheon-Sig Lee

coastalbend.edu/lee

Page 5

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