Math 417: Practice Problems for Section 1.1
These are additional practice problems for Section 1.1 in addition to Exercises 118 on page 5.
Find all solutions for each of the following linear systems of equations in three unknowns.
Systems with a unique solution

10x + 9y  z = 42 
18x + 12y  z = 80 
x  6y + z = 1 
Solution: (5, 2/3, 2)

x  z = 1 
6x + y z =12 
5x + y + z = 15 
Solution: (1, 8, 2)

x + 3y  z = 4 
x + 4y  z = 2 
4x  2y + z = 13 
Solution: (3, 0 ,1)

x  6y  z = 2 
5x  5y  z = 13 
8x + 7y + z = 11 
Solution: (2, 1, 2)

6x + 2y  z = 4 
10x + 3y z = 7 
x  y + z = 1 
Solution: (0, 3, 2)

2x + 6y + z = 41 
3x + 5y + z = 36 
2x + 7y + z = 47 
Solution: (1, 6, 3)

3x  14y  3z = 44 
6x  12y 3z = 21 
15x + 16y +3z = 73 
Solution: (2, 5/2, 1)

3x  y  z = 2 
10x  z = 17 
5x + 2y +z = 15 
Solution: (1, 8, 7)

3x  y  z = 2 
10x  z = 17 
5x + 2y +z = 15 
Solution: (1, 8, 7)

7x + 3y  z = 5 
12x + 4y  z = 5 
3x  2y + z = 6 
Solution: (1, 5, 3)
Systems with no solutions

3x + y + 27z = 2 
5x + 2y + 47z = 6 
7x + y + 55z = 5 

10x  3y  38z = 3 
7x  2y  26z = 2 
3x  3y  24z = 4 

8x + 3y  36z = 20 
11x + 4y  50z = 28 
3x + 3y  6z = 3 

6x + 7y 50 = 33 
7x + 8y  58z = 38 
5x + 7y + 16z = 8 
Systems with infinitely many solutions

5x +2y  17z = 2 
7x + 3y 24z = 32 
x + 2y 5z = 14 
Solution: (2+3t, 6+t, t) for every real number t

25x  4y + 50z = 63 
31x + 5y 62z = 78 
x + 4y + 2z = 15 
Solution: (3+2t, 3, t) for every real number t

2x  y + 2z = 6 
2x + y + 3z = 2 
4x  4y + 3z = 16 
Solution: (25t/4, 2t/2, t) for every real number t

4x + y  z = 1 
3x  2y + 2z = 2 
3y  3z = 3 
Solution: (0, 1+t, t) for every real number t

3x  y + 3z = 1 
6x + 3y  5z = 4 
3x + 6y + 2z = 11 
Solution: (0, 1+t, t) for every real number t