WEEK 6-APPLICATIONS-OF-LINEAR-SYSTEMS

Applications of Linear Systems

Objectives

Start

Methodology in Problem Solving

Selected Problems

Moving Objects in Physics

Restrictions in Linear Programming

Methodology in Problem Solving

Typical Steps Taken for Solving a Problem

STEP

Try it by yourself:

Selected Problems

Movie Tickets Budget

You have $24 to spend at the movies. Adult tickets cost $6 each and child tickets cost $3 each. If you want to buy 4 tickets in total,

a) Write the situation as a system of equations. b) how many of each type can you buy?

Solution

Cristine and James' Age

The age of Cristine added with James' age is 20. Additionally twice the age of Cristine is the same as triple of James' age.

a) Write the situation as a system of equations. b) Find the age of each one.

Solution

Moving Objects in Physics

Every object moving at constant velocity is described by the linear equation:

x=x₀+vt Where: x: Position x₀: Initial Position v: velocity t: time

When x≥0 we can change position x by the distance d.

Meeting Point of Converging Cars

Two cars are traveling towards each other from two different cities.

• Car A leaves City A heading towards City B at a constant speed of 60 km/h.
• Car B leaves City B heading towards City A at a constant speed of 80 km/h.

The distance between City X and City Y is 280 km.

a) Formulate the situation as a system of equations. b) Plot the trajectory of both cars. c) Find the distance from city B at which both cars meet using the plot of b).

Solution

Try it by Yourself

Now, imagine two airplanes taking the same trajectories but in opposite directions:

• Plane A departs at 6pm from Houston, Texas towards Mexico City at a constant speed of 400 km/h.
• Plane B departs at 8pm from Mexico City to Houston Texas at a constant speed of 400 km/h.

The distance between Houston, TX and Mexico City is 1000 km.

🤔

a) Formulate the system of equations. b) Plot the trajectory of both airplanes. c) Find the distance from Houston, TX at which both airplanes meet (use graphical method). d) Find the time at which they meet each other.

07:00

Restrictions in Linear Programming

Optimizing Production in a Factory

A factory produces two products, A and B. Each unit of product A cost $2, while each unit of product B costs $1. The factory only has $100 available. In addition, each product A requires one hour to be done, while product B requires 3 hours. There is a maximum of 90 hours for availabilty of the production machine.

a) Write given information in a table format. b) Express information as a system of inequalities. c) Plot all of the inequalities in a cartesian plane.

Solution

Company's annual gala

Sarah is organizing her company's annual gala, and the venue charges a flat rate of $9000 for five hours. The venue also provides meals for each guest and charges $25 per plate for adults and $15 per plate for children if she has a minimum of 80 guests. If Sarah’s budget is $45,000.

a) Write given information in a table format. b) Express information as a system of inequalities. c) Plot all of the inequalities in a cartesian plane.

Solution

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Solution

Answer

Step 1

Step 2

Step 4

Step 3

c) 900 kmd) 2 h 15 min

a)

b)

Mexico city

Houston

400km/h

-400km/h

1000km

Step 3.1

Step 3.2

Step 3.3

Solution

Answer

Step 1

Step 2

Step 4

Step 3

c) 120 km

a)

b)

cityB

cityA

60km/h

-80km/h

280km

Step 3.1

Step 3.2

Step 3.3

Create a plan of solution

Using the methods we've already seen or others. E.g solving equations graphically or by substitution.

Write givens on your Own Words

Making a list/table/drawing

Read the problem

Identify givens and question.

Solution

Answer

Step 1

Step 2

Step 4

Step 3

b) 4 adult tickets, 0 child tickets.

a)

b)

Step 3.1

Step 3.2

Step 3.3

MA.912.AR.2: Write, solve and graph linear equations, functions and inequalities in one and two variables MA.912.AR.2.5 Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine domain constraints in terms of the context. MA.912.AR.9: Write and solve a system of two- and three-variable equations and inequalities that describe quantities or relationships MA.912.AR.9.6 Given a real-world context, represent constraints as systems of linear equations or inequalities. Interpret solutions to problems as viable or non-viable options. ELD.K12.ELL.MA.1 English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.

State restrictions/conditions mathematically

• Each relevant statement should be represented in the form of equation or inequality.
• Some verbs indicate equality/inequality sign

Solution

Step 1

Step 2

Step 3

a)

c)

Step 3.1

b)

Step 3.2

Step 3.3

Verify the solution

that you have obtained. Also verify that it makes sense.

Define the variable(s)

Expressing variables with letters. They're usually in question.

Solve the problem

Using your own strategy.

Solution

Answer

Step 1

Step 2

Step 5

Step 4

Step 3

b) 4 adult tickets, 0 child tickets.

a)

b)

Step 3.1

Step 3.2

Step 3.3

MATERIAL

It is highly advised to have:

• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).

Solution

Step 1

Step 2

Step 3

a)

c)

Step 3.1

b)

Step 3.2

Step 3.3