AU MAT 120 Systems of Linear Equations and Inequalities Discussion

For the discussion board assignment in each unit, you will complete the problem associated with the letter that you have been assigned by the instructor. Post the entire example/word problem you have been assigned from the textbook. Then fully explain how you would go about finding the solution. Please explain all steps in a way that a struggling classmate could understand and learn from your methods.

Professional communication is expected in all posts, which includes proper spelling and grammar, and providing source information when using outside resources.

Each unit Discussion Forum will be worth 5 points and will be graded on the following criteria:

• Problem is stated at the start of the post
• All work/explanation for assigned problem is shown
• Correct solution
• Proper grammar and spelling are used

This is the Letter assigned to me below and the entire question to that letter.

R. Section 3.4

Solve the system by the method of your choice. If there is no solution or if there are infinitely manysolutions and a system’s equations are dependent, so state.

{ 3𝑥 − 5𝑦 = 7

𝑥 − 𝑦 = −1

BOTH EQUATION ARE IN THE BRACKET COULD’NT MAKE IT FIT BUT I ALSO ATTACH THE SHEET WITH THE PROBLEM AGAIN IT IS LETTER R

MAT120 – College Algebra
Unit 3 DB Assignment
A. Section 3.1
Explain how to solve a system of equations using the substitution method.
Use 𝑦 = 3 − 3𝑥 and 3𝑥 + 4𝑦 = 6 to illustrate your explanation.
B. Section 3.2
There were 180 people at a civic club fundraiser. Members paid $4.50 per ticket and nonmembers paid
$8.25 per ticket. If total receipts amounted to $1222.50, how many members and how many
nonmembers attended the fundraiser?
C. Section 3.4
In the following Exercise, perform the matrix row operation and write the new matrix.
1
−2 3 −10
[
|
] − 𝑅1
4 2 5
2
D. Section 4.4
Systems of inequalities will be used to model three of the target heart rate ranges shown in the bar graph.
We begin with the target heart rate range for cardiovascular conditioning, modeled by the following
system of inequalities:
10 ≤ 𝑎 ≤ 70
𝐻 ≥ 0.7(220 − 𝑎)
𝐻 ≤ .8(220 − 𝑎)
Heart rate ranges apply to
ages 10 through 70,
inclusive.
Target heart rate range is greater
than or equal to 70% of
maximum heart rate.
and less than or equal to 80% of
maximum heart rate.
The graph of this system is shown in the figure. Use the graph to solve the following Exercise.
a. What are the coordinates of point B and what does this mean in terms of age and heart rate?
b. Show that point B is a solution of the system of inequalities.
E. Section 4.5
Find the value of the objective function at each corner of the graphed region. What is the maximum
value of the objective function? What is the minimum value of the objective function?
Objective Function
𝑧 = 3𝑥 + 2𝑦
F. Section 3.1
Explain how to solve a system of equations using the addition method.
Use 5𝑥 + 8𝑦 = −1 and 3𝑥 + 𝑦 = 7 to illustrate your explanation.
G. Section 3.2
You invested $7000 in two accounts paying 6% and 8% annual interest. If the total interest earned for
the year was $520, how much was invested at each rate?
H. Section 3.4
In the following Exercise, perform the matrix row operation and write the new matrix.
1 −3 1
[
| ] − 2𝑅1 + 𝑅2
2 1 −5
I. Section 4.4
How do you determine if an ordered pair is a solution of an inequality in two variables, x and y?
J. Section 4.5
Find the value of the objective function at each corner of the graphed region. What is the maximum
value of the objective function? What is the minimum value of the objective function?
Objective Function
𝑧 = 30𝑥 + 45𝑦
K. Section 3.1
Solve the system by the addition method. Identify inconsistent systems and systems with dependent
equations, using set notation to express their solution sets.
{
𝑥 + 2𝑦 = −1
2𝑥 − 𝑦 = 3
L. Section 3.2
You invested money in two funds. Last year, the first fund paid a dividend of 8% and the second a
dividend of 5%, and you received a total of $1330. This year, the first fund paid a 12% dividend and the
second only 2%, and you received a total of $1500. How much money did you invest in each fund?
M. Section 3.4
Solve the system by the method of your choice. If there is no solution or if there are infinitely many
solutions and a system’s equations are dependent, so state.
{
𝑥 + 2𝑦 = 11

𝑥 − 𝑦 = −1

N. Section 4.4
 What does a solid line mean in the graph of an inequality?
 What does a dashed line mean in the graph of an inequality?
O. Section 4.5
A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $125 for
the rear-projection televisions and $200 for the plasma televisions.
a. Let 𝑥 = the number of rear-projection televisions manufactured in a month and 𝑦 = the number
of plasma televisions manufactured in a month. Write the objective function that models the
total monthly profit.
b. The manufacturer is bound by the following constraints:
 Equipment in the factory allows for making at most 450 rear-projection televisions in one
month.
 Equipment in the factory allows for making at most 200 plasma televisions in one month.

The cost to the manufacturer per unit is $600 for the rear-projection televisions and $900 for
the plasma televisions. Total monthly costs cannot exceed $360,000.
Write a system of three inequalities that models these constraints.
c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because x and
y must both be nonnegative.
d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed
region. [The vertices should occur at (0,0), (0,200), (300,200), (450,100), and (450,0).]
e. Complete the missing portions of this statement: the television manufacturer will make the greatest
profit by manufacturing ___ rear-projection televisions each month and ___ plasma televisions each
month. The maximum profit is $______.
P. Section 3.1
Solve the system by the method of your choice. Identify inconsistent systems and systems with
dependent equations, using set notation to express solution sets.
𝑦 = 3𝑥 + 5
{
5𝑥 − 2𝑦 = −7
Q. Section 3.2
When a small plane flies with the wind, it can travel 800 miles in 5 hours. When the plane flies in the
opposite direction, against the wind, it takes 8 hours to fly the same distance. Find the rate of the plane
in still air and the rate of the wind.
R. Section 3.4
Solve the system by the method of your choice. If there is no solution or if there are infinitely many
solutions and a system’s equations are dependent, so state.
{
3𝑥 − 5𝑦 = 7
𝑥 − 𝑦 = −1
S. Section 4.4
Explain how to graph 𝑥 − 2𝑦 < 4. T. Section 4.5 a. A student earns $15 per hour for tutoring and $10 per hour as a teacher’s aide. Let 𝑥 = the number of hours each week spent tutoring and 𝑦 = the number of hours each week spent as a teacher’s aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints:  To have enough time for studies, the student can work no more than 20 hours a week.  The tutoring center requires that each tutor spend at least three hours a week tutoring.  The tutoring center requires that each tutor spend no more than eight hours a week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because x and y are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at (3, 0), (8, 0), (3, 17), and (8, 12).] e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for _ hours per week and working as a teacher’s aide for __ hours per week. The maximum amount that the student can earn each week is $___. U. Section 3.1 When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations? V. Section 3.2 A company that manufactures small canoes has a fixed cost of $18,000. It costs $20 to produce each canoe. The selling price is $80 per canoe. (In solving this exercise, let x represent the number of canoes produced and sold.) For this exercise, a. Write the cost function, C. b. Write the revenue function, R. c. Determine the break-even point. Describe what this means. W. Section 3.4 In the following Exercise, perform the matrix row operation and write the new matrix. 1 [ 2 −3 5 | ] − 2𝑅1 + 𝑅2 6 4 X. Section 4.4  What is a solution of a system of linear inequalities?  Explain how to graph the solution set of a system of inequalities. Y. Section 4.5 Describe a situation in your life in which you would like to maximize something, but are limited by at least two constraints. Can linear programming be used in this situation? Explain your answer. Z. Section 3.1 When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?