Download Math 124 Unit 2 Homework

Math 124 Unit 2 Homework
1. Solve the following systems of equations by the graphing methods. Indicate at least two
points used for each line. ( 8 pts)
a.
5x – 3y = 9
x + 2y = 7
b. 3x - y = 4
6x – 2y = 12
Points used Line 1 ______________
Points used Line 1 ______________
Points used Line 2 ______________
Points used Line 2 ______________
Solution _____________
Solution __________________
2. Solve the following system of equations using substitution. If there is a specific solution,
state that solution. Otherwise, state no solution or many solutions. (2pts)
a. – x + 3y = 2
2x – 6y = -4
b. x + 4y = -1
2x – 5y = 11
3. Solve the following system of equations using the elimination method. If there is a specific
solution, state that solution. Otherwise, state no solution or many solutions. (2pts)
a. 6x + 3y = 9
-8x -4y = -12
b. x – 3y = -7
3x + 2y = 23
4. A supplier for the electronic industry manufactures keyboards and screens for graphing
calculators at plants in Mexico and Taiwan. The hourly production rates at each plant are
provided in the following table. How many hours should each plant be operated to exactly fill
an order for 4000 keyboards and 4000 screens. (2pts)
Plant
Mexico
Taiwan
Keyboards
40
20
Screens
32
32
a. Write a system of equations to represent the given problem.
b. Solve the system of equations using either the substitution or the elimination method.
5. A bike shop has fixed costs of $5200 to operate their business. Each bike they sell costs
$145 to produce. Their bikes sell for $225 each. ( 3pts)
a. Write the cost equation.
b. Write the revenue equation.
c. How many bikes do they have to sell to break even?
6. Suppose that the Supply and Demand for a product is given by the following equations
p = 1.5q + 4.2
p = −2.5q + 15
Supply Equation
Demand Equation
where p is the price in dollars and q is the quantity in thousands. ( 6 pts)
a) Find the supply and demand (to the nearest unit) if the product is priced at $6.
Supply __________
Demand ______________
b) Find the supply and demand (to the nearest unit) if the product is priced at $10.
Supply _____________
Demand ________________
c) Find the equilibrium price and quantity.
Price ______________
Quantity ___________________
7. State the dimensions of the following matrices. (3pts)
a.
3
[4 ]
3
b. [ 1
3
4 ]
3
c. [4
1
2 5
5 6]
2 3
8. Write an augmented matrix to represent the following system of equations. (2pts)
a. -3x - 6y + 9z = 12
-x + 2y= -10
x – 8z = -2
b. 4(x -3y) = 9y + 6
8x = 9y + 6
9. Write the system of equations represented by the following augmented matrix. (2pts)
1 0 0 3 
a.  0 1 0 0.05 
 0 0 1 12 
 8 7 113

b. 
2 0 1 
3 

10. Perform the following row operations on the given matrix. (2pts)
12 0 6 3 
a.  2 8 4 18 –R3 + R2  R2
 2 1 7
0 
 2 4 10 
b. 
5 3 
1
1
2
R1R1
11. Write as an augmented matrix and then use row reduction to solve. Show all steps in the row
reduction and label each. ( 8 pts)
a. 2x + 4y = -7
x – 3y= 9
c. x + 3y – 2z = 5
4x – y + 3z = 7
2x – 7y + 7z = 4
b. x – 6y = 4
5x – 30y = 20
d. 2x – 4y + 12z = 20
-x + 3y + 5z = 15
3x – 7y + 7z = 5
12. Find a,b,c,d for the following. (1pt)
[
4
𝑏
] + [
8
𝑑
𝑎
𝑐
a ___________
−8
7
] = [
−3
1
3
]
2
b ______________
c _____________ d ______________
13. Multiply the following matrices. Show all work. (2pts)
a.
[
𝑥
−𝑎
𝑦
𝑎
]x[
𝑥
−𝑏
𝑏
]
𝑦
b.
[
𝑦
−𝑏
𝑏
𝑥
] x[
−𝑎
𝑦
𝑎
]
𝑥
14. Determine the product of each matrix multiplication and then, state whether the following
pairs of matrices are inverses of each other. Calculator may be used for multiplication. (4 pts)
10 −3 1
a. [
] [
3 −1 3
3
]
−10
11 2 −8 3
b. [ 4
1 −3] [0
−8 −1 6 4
−4
2
−5
2
1]
3
15. Find the inverse, if possible. Use the method discussed in class, not the formula using the
determinant. Each row operation should be clearly marked. ( 4pts)
a. [
1 2
]
3 5
1 3
b. b. [0 1
3 2
9
4]
3
16. Write each system as a matrix equation and solve by using the inverse matrix method. Show
all work for finding the inverse. This includes clearly marking each row operation. A
calculator can be used for the multiplication part. (4 pts)
a. 1x + 2y – 1z = 2
1x + 1y + 2z = 0
1x – 1y – 1z = 1
b. 1x + 3y =5
2x + 1y = 10