1. Use elementary operations to find the solution set to the given linear systems.
Given:
7x1+3x211x1+5x2=6=12
[71135∣∣612]
Work:
Multiply top row by 3, and the bottom row by 2:
[2122910∣∣1824]
Swap the rows:
[2221109∣∣2418]
Subtract second row from first row and divide second row by 3:
[1713∣∣66]
Subtract 7* Row 1 from Row 2:
[101−4∣∣6−36]
Divide row 2 by -4:
[1011∣∣69]
Answer:
x2=9, and x1+9=6, so x1=−3
Given:
x1+2x2−x33x1−3x37x1+15x2+x3=−3,=−15,=4.
1372015−1−31∣∣∣−3−154
Work:
R2=R2−2R1:
1072−615−101∣∣∣−364
R3=R3−7R1:
1002−61−108∣∣∣−3625
R2=−6R2:
100211−108∣∣∣−3−125
R3=R3−R2:
100210−108∣∣∣−3−126
R3=8R3:
100210−101∣∣∣−3−1413
Answer:
x3=413, x2=−1, x1−2−213=−3, so x1=211
2. Write the augmented matrices for the following linear systems.
5x1+3x2−3x1−2x2=1,=4.→[5−33−2∣∣14]
2x1+3x24x1+7x33x2−x3=−1,=4,=0.→24030307−1∣∣∣−140
100010001∣∣∣−1−30
Matrix 1 is in reduced echelon form
010301−103∣∣∣041
Matrix 2 is not in reduced or echelon form
1000−11000210∣∣∣∣4−150
Matrix 3 is in reduced echelon form
4. Use Gaussian elimination to find the solution set to the following linear systems.
Given:
8x1+6x29x1+7x2=−4,=2.
[8967∣∣−42]
Work:
R1=R2−R1, R2=R1:
[1816∣∣6−4]
R2=(R2−8R1)/−2:
[1011∣∣626]
Answer:
x2=26 and x1+26=6, so x1=−20
Given:
2x1+3x24x1+7x33x2−x3=−1,=4,=0.
24030307−1∣∣∣−140
Work:
R1=R1−R3:
24000317−1∣∣∣−140
R2=R2−2R1
20000315−1∣∣∣−160
Swap R2 and R3:
2000301−15∣∣∣−106
Divide all the shits:
1000100.5−1/31∣∣∣−0.506/5
Answer:
x3=6/5
x2−(1/3)(6/5)=0 so x2=2/5
x1+(1/5)=−0.5 so x1=−7/10
5. Use Gauss–Jordan elimination to find the solution set to the following linear systems.
Given:
5x1+3x2−3x1−2x2=1=4
[5−33−2∣∣14]
Work:
R1=2R1+3R2:
[1−32−2∣∣144]
R2=R2+3R1:
[1024∣∣1432]
R2=R2/4:
[1021∣∣148]
Answer:
x2=8
x1=−2
Given:
2x1+3x2−x34x1+5x2−3x3−x1+3x2+5x3=−2=−2=−8
24−1353−1−35∣∣∣−2−2−8
Work:
R1=R1+R3:
14−16534−35∣∣∣−10−2−8
R2=R2+4R3:
10−161734175∣∣∣−10−26−8
R2=R2/17:
10−1613415∣∣∣−10−26/17−8
R3=(R3+R1)/9:
100611411∣∣∣−10−26/17−2
Answer:
The matrix is inconsistent because −26/17 can’t be equal to −2
Given:
2x1+3x2−x34x1+5x2−3x3−x1+3x2+5x3=4=−4=7
24−1353−1−35∣∣∣4−47
Work:
R2=R2−2R1:
20−13−13−1−55∣∣∣4−127
R1=−1(R1+R3):
10−1613455∣∣∣11127
R3=(R3−R1)/9:
100611451∣∣∣11122
R3=(R3−R2)/−4:
100610451∣∣∣11125/2
Answer:
x3=5/2
x2=−1/2
x1=4