Linear Systems and Problem Solving

Graphing – can provide a useful method for estimating a solution and to provide a visual model of the problem.

Substitution – requires that one of the variables be isolated on one side of the equation. It is especially convenient when one of the variables has a coefficient of 1 or –1.

Elimination Using Addition –convenient when a variable appears in different equations with coefficients that are opposites.

Elimination Using Subtraction –convenient if one of the variables has the same coefficient in the two equations.

Elimination Using Multiplication –can be applied to create opposites in any system.

Ways to Solve a System of Linear Equations

1) Write two sets of labels, if necessary (one set for number, one set for value, weight etc.)

2) Write two verbal models. (Translate from sentences)

3) Write two algebraic models (equations).

4) Solve the linear system.

5) Write a sentence and check your solution in the word problem.

Solving Word Problems Using A Linear System

Let m = Meg’s age

Meg’s age is 5 times Jose’s age. The sum of their ages is 18. How old is each person?

m =

Let j = Jose’s age

Assign Labels. Choose a different variable for each person.

Write an equation for each of the first two sentences.

m + j = 18

Solve the system of equations.18j) (

3j18j6

Sentence.

Jose is 3 and Meg is 15.

How old is Meg?

j5

15

35j5m

5j

110w2) (2

The length of a rectangle is 1 m more than twice its width. If the perimeter is 110 m, find the dimensions.

let w = widthlet l = length

Formula

length

length

widthwidth

l

18w108w61102w6110w22w4

The width is 18 m and the length is 37 m.

37136

1182

=

1w2 w22l 1w2

110

Example 1 A class has a total of 25 students. Twice the number of girls is equal to 3 times the number of boys. How many boys and girls are there in the class?

Assign Labels. Choose a different variable for each type of person. Let g = # of girls

g + b = 25

Let b = # of boysWrite an equation for each of the first two sentences.

2g b3 2

b10b550b350b2

There are 15 girls and 10 boys in the class.

25b g 25b =3b

15g2510g25bg

Example 2 The length of a rectangle is 4 m more than twice its width. If the perimeter is 38 m, find the dimensions.

5w30w6388w638w28w4

4. Solve the system.

1. Labels. let w = width let l = length

2. Translate first sentence.

3. Use perimeter formula.

length

length

widthwidth

l 2w2l

38w 2 2 and 4w2 ll

38w2) (2 5. Sentence.

The width is 5 m and the length is 14 m.

14410

452

=

4w2 4w2 l

42w

38

let a = # of adult tickets

Example 3 Admission to the play was $2 for an adult and $1.50 for a student. Total income from the sale of tickets was $550. The number of adult tickets sold was 100 less than 3 times the number of student tickets. How many tickets of each type were sold?

let s = # of student tickets

Number Labels.

Value Labels. let 2a =

let 1.50s =

value of adult tickets

value of student tickets

let a = # of adult tickets

Example 3 Admission to the play was $2 for an adult and $1.50 for a student. Total income from the sale of tickets was $550. The number of adult tickets sold was 100 less than 3 times the number of student tickets. How many tickets of each type were sold?

let s = # of student tickets

Number Labels.

Value Labels. let 2a =

let 1.50s =

value of adult tickets

value of student tickets

=

a =3s – 100 2a +1.50s=550 Clear the decimals. Multiply both sides by 100.55000s150a200

55000s150 200

100s3

100s

000,75s750

000,55000,20s750

000,55s150000,20s600

200

100300

1001003a

The school sold 200 adult tickets and 100 student tickets.

let q = # of quarters

Example 4 The number of quarters that Tom has is 3 times the number of nickels. He has $1.60 in all. How many coins of each type does he have?

let n = # of nickels

Number Labels. Value Labels. let .25q = value of

quarterslet .05n = value of nickels

let q = # of quarters

Example 4 The number of quarters that Tom has is 3 times the number of nickels. He has $1.60 in all. How many coins of each type does he have?

let n = # of nickels

Number Labels. Value Labels. let .25q = value of

quarterslet .05n = value of nickels

=

q =3n .25q +.05n= 1.60

Clear the decimals. Multiply both sides by 100.160n5q25

160n5 25

n3

2n160n80160n5n75

6

23q

Tom has 6 quarters and 2 nickels.

Example 5 The sum of two numbers is 100. Five times the smaller number is 8 more than the larger number. What are the two numbers?Assign Labels. Let s = smaller #

s + l = 100

Let l = larger #

Equations. 5s

8 5 l

lll

ll

826492

8650085005

The larger number is 82 and the smaller number is 18.

100 s l 100 l

=l + 8

Example 6 One number is 12 more than half another number. The two numbers have a sum of 60. Find the numbers.

Assign Labels. Let x = first # Let y = second #

Equations.

One number is 28 and the other number is 32.

12y21

x

60y12y21

60yx

6012y21

1

84y23

32

32

23y

12 12

Example 7 If you buy six pens and one mechanical pencil, you’ll get $1 change from your $10 bill. But if you buy four pens and two mechanical pencils, you’ll get $2 change. How much does each pen and pencil cost? Assign Labels. Let p =

pens6p + m = 10 - 1

Let m = mechanical pencils

Equations. 4p + 2m 8 2p4

Pens cost $1.25 each and mechanical pencils cost $1.50 each.

9p6m 9p6 =10 - 2

8 182p1p4

8 818p

10p8 8 8

45

p

18 18

25.1p

6p + m = 10 - 1

9m25.16

9m50.7 50.7 50.7

50.1m

SOLVE THE WORD PROBLEM:

An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How many five point questions are on the test?

• DEFINE THE VARIABLES:Let x = the number of 2 point questions and

y = the number of 5 point questions.

• WRITE A SYSTEM OF EQUATIONS:

An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How

many five point questions are on the test?

x+y=502x+5y=145

• SOLVE FOR ONE VARIABLE: x = 35 two-point questions

• SOLVE FOR THE OTHER VARIABLE:

x + y = 50 35 + y = 50 y = 15 five-point questions


• CHECK THE SOLUTION: (35, 15)

2(35) + 5(15) = 145 70 + 75 = 145

145 = 145

35 + 15 = 50 50 = 50



The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make?

• DEFINE THE VARIABLES:Let x = the number of 2 point baskets and

y = the number of 3 point baskets.


The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers

make? How many three-point shots did the Lakers make?

x+y=372x+3y=80

• SOLVE FOR ONE VARIABLE:

x = 31 two-point shots

• SOLVE FOR THE OTHER VARIABLE: x + y = 37 31 + y = 37

y = 6 three-point shots




2(31) + 3(6) = 80 62 + 18 = 80

80 = 80

x + y = 37 31 + 6 = 37

37 = 37




Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?

• DEFINE THE VARIABLES:Let x = the number of 2 point problems and

y = the number of 4 point problem.


Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some

problems are worth 4 points. How many problems of each value are on the test?

354

x+y=2x+ y=100


x = 20 two-point problems


y = 15 three-point problems



• CHECK THE SOLUTION:(20, 15)

2(20) + 4(15) = 100 40 + 60 = 100

100 = 100

x + y = 35 20 + 15 = 35 35 = 35




You are selling tickets for a musical at your local community college. Student tickets cost $5 and general admission tickets cost $8. If you sell 500 tickets and collect $3475, how many student tickets and how many general admission?

• DEFINE THE VARIABLES:Let x = the number of student tickets and

y = the number of general tickets.


You are selling tickets for a musical at your local community college. Student tickets cost $5 and general admission tickets cost $8. If you sell 500 tickets and

collect $3475,how many student tickets and how many general admission?

5003475

x+y=5x+8y=


x = 175 student tickets


y = 325 general tickets




5(175) + 8(325) = 3475 875 + 2600 = 3475

3475 = 3475

x + y = 500 175 + 325 = 500

500 = 500




The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480 and the total amount of money collected was $2340. How many rolls of each kind of gift wrap were sold?

• DEFINE THE VARIABLES:Let x = the amount of solid rolls and

y = the amount of printed rolls.


The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480 and the total amount of money collected

was $2340. How many rolls of each kind of gift wrap were sold?

x + y = 4804x + 6y = 2340


x = 270 solid rolls

• SOLVE FOR THE OTHER VARIABLE: x + y = 480

270 + y = 480 y = 210 printed rolls




4(270) + 6(210) = 2340 1080 + 1260 = 2340

2340 = 2340

x + y = 480 270 + 210 = 480

480 = 480



Linear Systems and Problem Solving

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