Name: Chapter 3: Systems of Equations and Inequalities ...simonclps.weebly.com/uploads/2/2/6/9/22693662/a2_notes_3.pdf · Solve the system of inequalities by graphing. ... Lesson

Name: Chapter 3: Systems of Equations and Inequalities

Page 1

Lesson 3-1: Solving Systems of Equations Date:

A is two or more equations with the same variables.

To solve a system of equations with two variables, find the ordered pair that satisfies all of the equations

by using the following methods:

1.

2.

3.

Systems of equations can be classified by their number of solutions.

A system is if it has at least one solution.

o If a system has exactly one solution, it is called .

o If a system has an infinite number of solutions, it is called .

A system is if it has no solutions.

Example 1: Solve by Graphing

Solve the system of equations by graphing. Then describe it as consistent and independent, consistent and

dependent, or inconsistent.

A.

x

y


Page 2

B.

C.

D.

x

y

x

y

x

y


Page 3

Example 2: Use the Substitution Method

FURNITURE Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A

rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last

month, the business earned $13,930 for the 48 outdoor chairs sold. How many of each chair were sold?

Write the equations:

Step 1:

Step 2:

Step 3:


Page 4

Example 3: Solve by Using Elimination

Use the elimination method to solve the system of equations.


Page 5

Example 5: No Solution and Infinite Solutions

Solve the system of equations.


Page 6

Lesson 3-2: Solving Systems of Inequalities by Graphing Date:

Solving a system of means finding the ordered pairs that satisfy all of the

inequalities in the system.

Example 1: Intersecting Regions

Solve the system of inequalities by graphing.

Example 2: Separate Regions

Solve the system of inequalities by graphing.

x

y

x

y


Page 7

Real-World Example 3: Write and Use a System of Inequalities

MEDICINE Medical professionals recommend that patients have a cholesterol level c below 200

milligrams per deciliter (mg/dL) of blood and a triglyceride level t below 150 mg/dL. Write and graph a

system of inequalities that represents the range of cholesterol levels and triglyceride levels for patients.

Example 4: Find Vertices

Find the coordinates of the vertices of the triangle formed by 2𝑥 − 𝑦 ≥– 1, 𝑥 + 𝑦 ≤ 4, and 𝑥 + 4𝑦 ≥ 4.

x

y

x

y


Page 8

Lesson 3-3: Optimization with Linear Programming Date:

is a method for finding maximum or minimum values of a function

over a given system of inequalities with each inequality representing a constraint.

Example 1: Bounded Region

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region.

Find the maximum and minimum values of the function 𝑓(𝑥, 𝑦) = 3𝑥 – 2𝑦 for this region.

𝑥 ≤ 5

𝑦 ≤ 4

𝑥 + 𝑦 ≥ 2

Step 1: Graph the inequalities.

Step 2: Use a table to find the maximum and minimum

values of 𝑓(𝑥, 𝑦) by using the vertices from the graph.

(𝑥, 𝑦) 𝑓(𝑥, 𝑦)

x

y


Page 9

Example 2: Unbounded Region

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find

the maximum and minimum values of the function 𝑓(𝑥, 𝑦) = 2𝑥 + 3𝑦 for this region.

– 𝑥 + 2𝑦 ≤ 2

𝑥 – 2𝑦 ≤ 4

𝑥 + 𝑦 ≥ – 2

Step 1: Graph the inequalities

Step 2: Use a table to find the maximum and minimum values of 𝑓(𝑥, 𝑦) by using the vertices from the

graph.

(𝑥, 𝑦) 𝑓(𝑥, 𝑦)

x

y


Page 10

To means to seek the best price or amount to minimize costs or

maximize profits.

Real-World Example 3: Optimization with Linear Programming

LANDSCAPING A landscaping company has crews who mow lawns and prune shrubbery. The company

schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2

pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the

charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of

mowing lawns and pruning shrubs that will maximize the income the company receives per day from one of

its crews.

(𝑥, 𝑦) 𝑓(𝑥, 𝑦)

x

y


Page 11

Lesson 3-4: Systems of Equations in Three Variables Date:

Like systems of equations in two variables, systems in three variables can have one solution, infinite

solutions, or no solution.

A solution of such a system is called an (𝑥, 𝑦, 𝑧).

Example 1: A System with One Solution



Page 12

Example 2: No Solution and Infinite Solutions


A.

B.


Page 13

Real-World Example 3: Write and Solve a System of Equations

SPORTS There are 49,000 seats in a sports stadium. Tickets for the seats in the upper level sell for $25, the

ones in the middle level cost $30, and the ones in the bottom level are $35 each. The number of seats in the

middle and bottom levels together equals the number of seats in the upper level. When all of the seats are

sold for an event, the total revenue is $1,419,500. How many seats are there in each level?


Page 14

Lesson 3-5: Operations with Matrices Date:

Real-World Example 1: Analyze Data with Matrices

Use the matrix below that includes information on tuition (T), room and board (R/B), and enrollment (E)

for three universities.

𝑇 𝑅/𝐵 𝐸𝑀𝑆𝑈 6160 5958 26160𝑈𝑀 6293 7250 30409

𝐶𝑀𝑈 5352 6280 12609

A. Find the average of the elements in column 1, and interpret the result.

B. Which university’s total cost is the lowest?

C. Would adding the elements of the rows provide meaningful data? Explain.

D. Would adding the elements of the third column provide meaningful data? Explain.


Page 15

Example 2: Add and Subtract Matrices

A. Find 𝐴 + 𝐵 if 𝐴 = [6 4

−1 0] and 𝐵 = [

−3 10 3

].

B. Find 𝐴 − 𝐵 if 𝐴 = [4 −2 01 5 −1

] and 𝐵 = [−6 7−9 3

].


Page 16

Example 3: Multiply a Matrix by a Scalar

If 𝐴 = [2 1

−1 30 5

], find 2𝐴.


Page 17

Example 4: Multi-Step Operations

If 𝐴 = [2 3

−1 0] and 𝐵 = [

−2 10 −1

], find 4𝐴 − 3𝐵.

Real-World Example 5: Use Multi-Step Operations with Matrices

BUSINESS A small company makes unfinished desks and cabinets. Each item requires different amounts

of hardware as shown in the matrices.

The company has orders for 3 desks and 4 cabinets. Express the company’s total needs for hardware in a

single matrix.


Page 18

Lesson 3-6: Multiplying Matrices Date:

You can multiply two matrices A and B if and only if the number of columns in A is equal to the number

of rows in B.

Examples:

Example 1: Dimensions of Matrix Products

Determine whether the product of the two matrices is defined. If so, state the dimensions of the product.

A. 𝐴3×4 and 𝐵4×2 B. 𝐴3×2 and 𝐵4×3

Example 2: Multiply Square Matrices

Find 𝑅𝑆 if 𝑅 = [3 2

−1 0] and 𝑆 = [

−2 11 −1

]


Page 19

Real-World Example 3: Multiply Matrices

CHESS Three teams competed in the final round of the Chess Club’s championships. For each win, a team

was awarded 3 points and for each draw a team received 1 point. Which team won the tournament?

Example 4: Test of the Commutative Property

A. Find 𝐾𝐿 if 𝐾 = [−3 2 2−1 −2 0

] and 𝐿 = [1 −24 30 −1

]

B. Find 𝐿𝐾 if 𝐾 = [−3 2 2−1 −2 0

] and 𝐿 = [1 −24 30 −1

]


Page 20

Example 5: Test of the Distributive Property

A. Find 𝐴(𝐵 + 𝐶) if 𝐴 = [−1 22 1

] and 𝐵 = [1 03 −2

] and 𝐶 = [−3 1−1 0

]

B. Find 𝐴𝐵 + 𝐴𝐶 if 𝐴 = [−1 22 1

] and 𝐵 = [1 03 −2

] and 𝐶 = [−3 1−1 0

]


Page 21

Lesson 3-7: Solving Systems of Equations Using Cramer’s Rule Date:

Every square matrix has a .

The determinant of a 2 × 2 is called a determinant.

Example 1: Second-Order Determinant

Evaluate the determinant |6 4

−1 0|


Page 22

Determinant of 3 × 3 matrices are called determinants.

They can be evaluated using the .

Example 2: Use Diagonals

Evaluate |3 −2 −12 −1 01 2 −3

| using diagonals.


Page 23

Real-World Example 3: Use Determinants

SURVEYING A surveying crew located three points on a map that formed the vertices of a triangular area.

A coordinate grid in which one unit equals 10 miles is placed over the map so that the vertices are located at

(0, –1), (–2, –6), and (3, –2). Use a determinant to find the area of the triangle.


Page 24

Determinants are used to solve systems of equations.

If a determinant is nonzero, then the system has a unique solution.

If a determinant is zero, then the system either has no solution or infinite solutions.

Cramer’s Rule uses the matrix, which is a matrix that contains only the

coefficients of a system.

Example 4: Solve a System of Two Equations

Use Cramer’s Rule to solve the system of equations.

5𝑥 + 4𝑦 = 28

3𝑥 − 2𝑦 = 8


Page 25

Example 5: Solve a System of Three Equations

Solve the system by using Cramer’s Rule.

2𝑥 + 𝑦 − 𝑧 = −2

−𝑥 + 2𝑦 + 𝑧 = −0.5

𝑥 + 𝑦 + 2𝑧 = 3.5


Page 26

Lesson 3-8: Solving Systems of Equations Using Inverse Matrices Date:

A matrix is a matrix with the same number of rows and columns.

The matrix is a square matrix that, when multiplied by another matrix, equals

that same matrix.

Two 𝑛 × 𝑛 matrices are of each other if their products is the identity matrix.

Example 1: Verify Inverse Matrices

A. Determine whether 𝑋 and 𝑌 are inverses.

𝑋 = [3 −2

−1 1] and 𝑌 = [

1 21 3

]

B. Determine whether 𝑃 and 𝑄 are inverses.

𝑃 = [3 −14 −2

] and 𝑄 = [1 −32 4

]


Page 27

You can determine whether a matrix has an inverse by using the determinant. (if the value of the

determinant is zero, the matrix cannot have an inverse)

Example 2: Find the Inverse of a Matrix

Find the inverse of the matrix if it exists.

A. 𝑆 = [−1 08 −2

]

B. 𝑇 = [−4 6−2 3

]


Page 28

Real-World Example 3: RENTAL COSTS The Booster Club for North High School plans a picnic. The

rental company charges $15 to rent a popcorn machine and $18 to rent a water cooler. The club spends

$261 for a total of 15 items. How many of each do they rent?

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