Assignment
1. Use the graphing approach to determine whether the system is consistent, the system is inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. (If the system is dependent, enter DEPENDENT. If the system is inconsistent, enter INCONSISTENT.)
x - y = 1
2x + y = 8
2. Solve the system by using either the substitution method or the eliminationbyaddition method, whichever seems more appropriate. (If the system is dependent, enter DEPENDENT. If the system is inconsistent, enter INCONSISTENT.)
5x - y = -43
2x + 3y = -7
3. Solve the system by using either the substitution method or the eliminationbyaddition method, whichever seems more appropriate. (If the system is dependent, enter DEPENDENT. If the system is inconsistent, enter INCONSISTENT.)
y = 4x - 20
7x + y = 35
4. Solve the problem by using a system of equations.
A motel rents double rooms at $34 per day and single rooms at $23 per day. If 28 rooms were rented one day for a total of $831, how many rooms of each kind were rented?
5. Solve the problem by using a system of equations.
Sam invested $1750, part of it at 6% and the rest at 8% yearly interest. The yearly income on the 8% investment was $30 more than twice the income from the 6% investment. How much did he invest at each rate?
6. Solve the system.
x - 2y + 3z = 16
2x + y + 5z = 34
3x - 4y - 2z = 17
7. Solve the problem by setting up and solving a system of three linear equations in three variables.
A box contains $8.00 in nickels, dimes, and quarters. There are 47 coins in all, and the sum of the numbers of nickels and dimes is 3 less than the number of quarters. How many coins of each kind are there?
8. Indicate whether the matrix is in reduced echelon form.
9. Indicate whether the matrix is in reduced echelon form.
10. Indicate whether the matrix is in reduced echelon form (x1, x2, x3, x4).
x1 - 3x2 - 2x3 + x4 = -3
-2x1 + 7x2 + x3 - 2x4 = -1
3x1 - 7x2 - 3x3 + 3x4 = -5
5x1 + x2 + 4x3 - 2x4 = 18
11. The matrix is the reduced echelon matrix for a system with variables x1, x2, x3, and x4. Find the solution set of the system. (If the system has infinitely many solutions, express your answer in terms of k, where x1 = x1(k), x2 = x2(k), x3 = x3(k), and x4 = k. If the system is inconsistent, enter INCONSISTENT.)
12. Use the following definition.
Evaluate the 2 × 2 determinant.
13. Evaluate the 3 × 3 determinant. Use the properties of determinants to your advantage.
14. Use Cramer's rule to find the solution set for the system. (If the system is dependent, enter DEPENDENT. If the system is inconsistent, enter INCONSISTENT.)
2x - y = -5
3x + 2y = 17
15. Use Cramer's rule to find the solution set for the system. (If the system is dependent, enter DEPENDENT. If the system is inconsistent, enter INCONSISTENT.)
3x - 2y - 3z = -12
x + 2y + 3z = -4
-x + 4y - 6z = 8
16. Indicate the solution set for the system of inequalities by graphing the system and shading the appropriate region.
x + y > 1
x - y > 4
17. Indicate the solution set for the system of inequalities by graphing the system and shading the
appropriate region.
x ≤ 3
y ≤ -6
18. Find the maximum value and the minimum value of the given function in the indicated region.
f(x, y) = 8x + 3y
19. Maximize the function
f(x, y) = 4x + 9y
in the region determined by the following constraints.
3x + 2y ≤ 18
3x + 4y ≥ 12
x ≥ 0
y ≥ 0
20. Solve the linear programming problem by using the graphing method illustrated in this example.
A manufacturer of golf clubs makes a profit of $45 per set on a model A set and $40 per set on a model B set. Daily production of the model A clubs is between 30 and 60 sets, inclusive, and that of the model B clubs is between 10 and 30 sets, inclusive. The total daily production is not to exceed 60 sets. How many sets of each model should be manufactured per day to maximize the profit?
model A sets
model B sets