Chapter 4Systems of Equations

4.1 Systems of Equations in Two Variables

A system of two linear equations in two variables x and y consists of two equations, Ax + By = C and Dx + Ey = FA solution of a system of linear equations in two variables is an ordered pair (x, y) that satisfies both equations.

NUMBER OF SOLUTIONS OF A LINEAR SYSTEMCONCEPT

SUMMARY

y

x

y

x

Lines intersectone solution

Lines are parallelno solution

y

x

Lines coincideinfinitely many solutions

Check, by graphing, whether each system of 3 equations has a common solution. If it does, give the solution. If it does not, state that it does not.

Check, by graphing, whether each system of 3 equations has a common solution. If it does, give the solution. If it does not, state that it does not.

Without graphing, tell whether or not the following system, in which a b, has a solution, and if so what it is. If it does not have a solution, explain why not.

Explain why your answer above does not depend on the values of a and b, as long as a b.

A: Put each of the equations in the following system into slope intercept form.

B: From your answer to A, and without graphing, tell whether the graphs of the two equations intersect once, do not intersect, or define the same line.

C: Based on your answer to part B, describe a general method for determining the nature of the solution(s) of a linear system without graphing.

HW #4.1Pg 161 1-15 Odd, 16-18

Pg 161 5 Pg 161 7 Pg 161 16 Pg 161 18

Pg 161 9 Pg 161 11 Pg 161 15d Pg 161 18

HW Quiz 4.1Tuesday, April 18, 2023

4.2 Solving Systems of Equations

Objective: Solve systems of linear equations by substitution

Objective: Solve systems of linear equations by substitution

A B

Objective: Solve systems of linear equations by Linear Combinations

Objective: Solve systems of linear equations by Linear Combinations

C D

4.3 Using Systems of Equations

You and a friend share the driving on a 280 mile trip. Your average speed is 58 miles per hour. You friend’s average speed is 53 miles per hour. You drive one hour longer than your friend. How many hours did each of you drive?

A gardener has two solutions of weedkiller and water. One is 5% weedkiller and the other is 15% weedkiller. The gardener needs 100 L of a solution that is 12% weedkiller. How much of each solution should she use?

A freight train leaves Tyler, traveling east at 35 km/h. One hour later a passenger train leaves Tyler, also traveling east on a parallel track at 40 km/h. How far from Tyler will the passenger train catch the freight train?

E The band boosters are organizing a trip to a national competition for the 226-member marching band. A bus will hold 70 students and their instruments. A van will hold 8 students and their instruments. A bus costs $280 to rent for the trip. A van costs $70 to rent for the trip. The boosters have $980 to use for transportation. Write a system of equations whose solution is how many buses and vans should be rented. Solve the system.

HW #4.2-3Pg 166-167 1-25 odd, 26-29

Pg 173 27-33 Odd, 34

Pg 167 28 Pg 167 29 Pg 173 31 Pg 173 34

Pg 167 28 Pg 167 29 Pg 173 34 Pg 173 31

HW Quiz #4.2-3Tuesday, April 18, 2023

4.4 Systems of Equations in Three Variables

An Equation with Three Variables

• A solution of a system of equations in three variables is called an ordered triple.

• What is an ordered triple? – ( x, y, z) An example is (2 , -4, 3).

• This ordered triple must be true for all the equations in the system.

• The graph of a linear equation in three variables is a plane. Thus, if a system of equations in three variables has a unique solution, it is a point common to all of the planes.

How should we solve a system in three variables??

We will first solve this equation:

x + y + z = 4

x - 2y - z = 1

2x- y -2z = -1

1

2

3

These numbers indicate the equations in the first, second, and third positions.

First, pick a variable to eliminate. We will eliminate x in this system.

Solving a system of three equations

x + y + z = 4

x - 2y - z = 1

2x- y -2z = -1

1

2

3

Step 3: -3y -2z = -3 -3y -4z = -9

Step 2: *(-2) -2x- 2y - 2z = -8 2x - y - 2z = -1 -3y - 4z = -9

1

3

Step 1: *(-1) -x- y - z = -4 x- 2y - z = 1 - 3y -2z = -3

1

2

Solving a system of three equations

continuedNext part of Step 3:

Now just substitute in 3 for z and find the other variables.

(-3y -2z = -3)*(-1) -3y -4z = -9

3y + 2z = 3 -3y - 4z =-9 -2z = -6 z = 3

The AnswersYou should have gotten:

x = 2

y = -1

z = 3

The ordered triple is (2, -1, 3)

Let’s do ANOTHER example!

Let’s solve this system of equations:

2x + y + 2z = 11

3x +2y + 2z = 8

x +4y + 3z = 0

1

2

3

Step 1: *(-2) -2x - 8y - 6z = 0

2x + y + 2z =11

-7y - 4z =11

3

1

Solving the system of equations continued

Step 2: *(-3) -3x - 12y - 9z = 0 3x + 2y + 2z = 8

-10y - 7z = 8

3

2

From Step 1Step 3: ( -7y - 4z = 11) * (-10) (-10y - 7z = 8)* (7) 70y +40z = -110 -70y - 49z = 56

-9z = -54 z = 6

Solving the system of equations continued

• Now plug in 6 for the variable z to find x and y.

• You should get:

x = 2y = -5

z = 6

The ordered triple is (2, -5, 6)

And yet ANOTHER example!

• But this one has a special twist!

• Systems of three equations can have infinite many solutions or no solutions.

• Here is an example with infinite many solutions:

x – 3y + z = 1 2x – y – 2z = 2 x + 2y – 3z = -1

1

2

3

Solving a system of equations that has infinite many solutions continued

Step 1: *(-3) -6x - 3y + 9z = -12 -2x +3y -13z = -8 -8x - 4z = -20

1

3

Step 2: *(1/2) -8x - 4z = -20 -4x - 2z = -10

Step 3: -4x - 2z = -10 4x + 2z = 10

0 = 0Signifies the system of equations has infinite many solutions

Summary• Remember, solutions are written as an ordered

triple.• Remember, solutions can also be no solution or

infinite many solutions. • If one equation is missing a variable, just line it up

with the other equations.For example: 3p + 2r = 11

q - 7r = 4 p - 6q = 1

• Check your solutions in all equations to make sure it is correct. Just because it answers two of them, doesn’t mean it answers all of them.

A B

HW #4.4a Pg 178-179 1-27 Odd, 28-30

4.4 Systems of Equations in Three Variables

Groups to work on solving

HW #4.4b Pg 178-179 2-26 Even

HW Quiz 4.4bTuesday, April 18, 2023

4.5 Using a System of Three Equations

In yesterday’s swim meet, Roosevelt High dominated in the individual events, with 24 individual-event placers scoring a total of 56 points. A first-place finish scores 5 points, a second-place finish scores 3 points, and a third-place finish scores 1 point. Having as many third-place finishers as first- and second-place finishers combined really shows the team’s depth. Use a system of three equations in 3 variables to determine the number of 1st, 2nd, and 3rd place finishers Roosevelt had.

You have $25 to spend on picking 21 pounds of three different types of apples in an orchard. The Empire apples cost $1.40 per pound, the Red Delicious apples cost $1.10 per pound, and the Golden Delicious apples cost $1.30 per pound. You want twice as many Red Delicious apples as the other two kinds combined. Write a system of equations to represent the given information. How many pounds of each type of apple should you buy?

Gina sells magazines part time. On Thursday, Friday, and Saturday, she sold $66 worth. On Thursday she sold $3 more than on Friday. On Saturday she sold $6 more than on Thursday. How much did she take in each day?

Find a three digit positive integer such that the sum of all three digits is 14, the tens digit is two more than the ones digit, and if the digits are reversed the number is unchanged.

HW #4.5Pg 181-182 1-17 Odd, 18-19

11 13 15 17

9 11 13 17

HW Quiz #4.5Tuesday, April 18, 2023

4.6 Consistent and Dependent Systems

Objective: Determine whether a system of equations is consistent or inconsistent.

Consistent System: If a system of equations has at least one solution

Inconsistent System: If a system of equations has no solution

Objective: Determine whether a system of equations is consistent or inconsistent.

Objective: Determine whether a system of equations is dependent

A. Is it possible to have a system that is consistent and dependent?

B. Is it possible to have a system that is inconsistent and dependent?

C. Is it possible to have an inconsistent system that is not dependent?

D. Is it possible to have an consistent system that is not dependent?

Objective: Determine whether a system of equations is dependent

Determine if the systems are consistent, inconsistent and dependent

A B

HW #4.6Pg 186-187 1-23 Odd, 25-31

HW Quiz 4.6Tuesday, April 18, 2023

4.7 Systems of Linear Inequalities

Objective: Graph a linear inequality

The boundary line of the inequality divides the coordinate plane into two half-planes; a shaded region containing the points that are solutions of the inequality, and an unshaded region which contains the points that are not.

Objective: Graph a linear inequality

Objective: Graph a system of linear inequalities

A system of linear inequalities is two or more linear inequalities in the same variables and is also called a system of inequalities.A solution of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system. The graph of a system of linear inequalities is the graph of all solutions of the system.

Objective: Graph a system of linear inequalities

Objective: Graph a system of linear inequalities

Objective: Graph a system of linear inequalities

Objective: Graph a system of linear inequalities

HW #4.7Pg 192 1-39 Odd, 40-41

4.8 Using Linear Programming

Objective: Solve problems using linear programming.

A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread they make $35 profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take 0.5 hour to prepare and 0.5 hour to bake. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should be made to maximize profits?

Objective: Solve problems using linear programming.

Optimization means finding the maximum or minimum value of some quantity.

Linear programming is the process of optimizing a linear objective function subject to a system of linear inequalities called constraints.

Objective: Solve problems using linear programming.

Objective: Solve problems using linear programming.

Objective: Solve problems using linear programming.

A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread they make $35 profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take 0.5 hour to prepare and 0.5 hour to bake. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should be made to maximize profits?

Objective: Solve problems using linear programming.

(16, 2)

(20, 0)

(0, 4)

(0, 0)

Muffins

Bre

ad

Objective: Solve problems using linear programming.

Wheels Inc. makes mopeds and bicycles. Experience shows they must produce at least 10 mopeds. The factory can produce at most 60 mopeds per month. The profit on a moped is $134 and on a bicycle, $20. They can make at most 120 units combined. How many of each should they make per month to maximize profit?

Objective: Solve problems using linear programming.

(110, 10)

(60, 60)(0, 60)

(0, 10)

Bikes

Mop

eds

Objective: Solve problems using linear programming.

Farm Management A farmer has 70 acres of land available for planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table:

The farmer cannot spend more than $1800 in preparation costs nor use more than a total of 120 workdays. How many acres of each crop should be planted to maximize the profit? What is the maximum profit?

(0. 30)

(24. 12)

(30. 0)(0. 0)

Soy

Whe

at

A manufacturer of skis produces two types: downhill and cross- country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit?

HW #4.8 Pg 195 1-6

HW Quiz 4.8Tuesday, April 18, 2023

Test Review

Part 1

Find a and b so that the system below has the unique solution (-2, 3)

Find p and q so that the graph of the equation y = x2 + px + q passes through (-1, 3) and (2, 4)

Know the terms: Dependent, Consistent, and Inconsistent

Part 2

Part two will consist of three linear programming problems along with one proof

HW # R-4Pg 200 1-13

Study All Challenge Problems

Find the area of an equilateral triangle