1. Determine which of the following points satisfy the system of equations given below.
Circle the letter corresponding to each point that satisfies the system.
(a) (2, 0, 7)
(b) (0, 2, 5)
(c) (1, 1, 7)
(d) (7, –1, 3)
2x + 4y + z = 13
x + 3y – z = 1
–3x + y + 4z = 22
2. Use only the elimination method to find the point of intersection of the following
pair of lines. Eliminate x to find y and then eliminate y to find x. DO NOT USE
SUBSTITUTION.
3x + 8y = 5
2x – 4y = –3
3. Use only the substitution method to find the point of intersection of the following
pair of lines. DO NOT USE ELIMINATION.
x + 3y = 7
5x + 6y = 8
4. Use the method of elimination and/or substitution to find the solution of the following
system of equations.
3x + 4y – 2z = 4
2x + y – 2z = –6
x – 2y + 3z = –1
5. Janet can purchase her favorite candy bars in packages of 4 candy bars or 6 candy
bars per package. On Saturday she bought 12 packages and ended up with a total of
62 candy bars. Let x be the number of 4-bar packages she bought and let y be the
number of 6-bar packages she bought. Set up and solve a system of equations to
determine x and y.
6. For each of the following systems of equations, determine the augmented matrix, the
reduced row echelon form of the augmented matrix, and the solution of the system. If
the system has no solution write NONE for the solution.
Augmented Matrix:
3x – 5y + 4z = 7
–x + 3y – z = –4
RREF:
2x + 2y + 4z = –2
Solution:
Augmented Matrix:
3x – 2y + z = 8
2x + 5y – z = –4
RREF:
5x
+ 4z = 2
Solution:
Augmented Matrix:
3x + y – 5z = 2
–2x + 3y + 2z = –3
RREF:
x + 15y – 7z = 5
Solution:
7. (a) What system of equations is represented by the following augmented matrix?
1 2 3 4 
0 1 2 5


0 0 1 2 
(b) Use back substitution to solve the system of equations in part (a).
 2 5
8. Let B = 
 . By hand, calculate the determinant of B.
4 3 
 5 2 3
9. Let C =  4 1 7  . Use your calculator to calculate the determinant of C.


 2 1 6 
10. Determine the partial fraction decomposition of
9 x 2  26 x  8
.
x 2 ( x  4)
11. Determine the partial fraction decomposition of
5 x  14
.
( x  2)( x 2  4)
12. Solve the following system of equations using Cramer’s rule.
3x – 5y = –4
2x + 2y = 5
D = __________
Solution:
Dx = ___________
____________________
Dy = ___________
13. Use a determinant to calculate the area of the triangle whose vertices are (5, 2),
(3, –2) and (7, 10).