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Linear Systems and Problem Solving

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Ways to Solve a System of Linear Equations

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

Substitution â€“ 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 â€“1.

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

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

Elimination Using Multiplication â€“can be applied to create opposites in any system.

Solving Word Problems Using A Linear System

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.

Megâ€™s age is 5 times Joseâ€™s age. The sum of their ages is 18. How old is each person?

Assign Labels. Choose a different variable for each person.

Let m = Megâ€™s age

Let j = Joseâ€™s age

Write an equation for each of the first two sentences.

m =

5j

m + j = 18

Solve the system of equations.

How old is Meg?

Sentence.

Jose is 3 and Meg is 15.

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

=

length

letl =length

let w = width

width

width

length

Formula

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

Example 1A 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

Let b = # of boys

Write an equation for each of the first two sentences.

g+b = 25

2g

3b

=

There are 15 girls and 10 boys in the class.

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

=

length

1. Labels.

letw= width

letl = length

width

width

length

2. Translate first sentence.

3. Use perimeter formula.

4. Solve the system.

5. Sentence.

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

Example 3Admission 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?

Number Labels.

let a = # of adult tickets

let s = # of student tickets

value of

adult tickets

value of student tickets

Value Labels.

let 2a =

let 1.50s =

Example 3Admission 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?

=

Number Labels.

let a = # of adult tickets

let s = # of student tickets

value of

adult tickets

value of student tickets

Value Labels.

let 2a =

let 1.50s =

Clear the decimals. Multiply both sides by 100.

=

a

3sâ€“ 100

1.50s

2a

+

550

=

The school sold 200 adult tickets and 100 student tickets.

Example 4The 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 q = # of quarters

let n = # of nickels

Number Labels.

Value Labels.

let .25q = value of quarters

let .05n = value of nickels

Example 4The 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 q = # of quarters

let n = # of nickels

Number Labels.

Value Labels.

let .25q = value of quarters

let .05n = value of nickels

=

q

Clear the decimals. Multiply both sides by 100.

3n

.25q

.05n

+

= 1.60

Tom has 6 quarters and 2 nickels.

Example 5The sum of two numbers is 100. Five times the smaller number is 8 more than the larger number. What are the two numbers?

Let l= larger #

Assign Labels.

Let s = smaller #

l+ 8

=

s + l= 100

5s

Equations.

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

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

Let y= second #

Assign Labels.

Let x = first #

Equations.

One number is 28 and the other number is 32.

Example 7If you buy six pens and one mechanical pencil, youâ€™ll get $1 change from your $10 bill. But if you buy four pens and two mechanical pencils, youâ€™ll get $2 change. How much does each pen and pencil cost?

Let m = mechanical pencils

Assign Labels.

Let p = pens

6p + m = 10 - 1

10 - 2

=

4p + 2m

6p + m = 10 - 1

Equations.

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

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?

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?

- 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 = 145ïƒ¼

35 + 15 = 50

50 = 50ïƒ¼

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?

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.

- WRITE A SYSTEM OF EQUATIONS:

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?

- 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

- CHECK THE SOLUTION:(31, 6)
2(31) + 3(6) = 80

62 + 18 = 80

80 = 80ïƒ¼

x + y = 37

31 + 6 = 37

37 = 37ïƒ¼

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

Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems.Â 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.

- WRITE A SYSTEM OF EQUATIONS:

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

- SOLVE FOR ONE VARIABLE:
x = 20 two-point problems

- SOLVE FOR THE OTHER VARIABLE:
x + y = 35

20 + y = 35

y = 15 three-point problems

- CHECK THE SOLUTION:(20, 15)
2(20) + 4(15) = 100

40 + 60 = 100

100 = 100ïƒ¼

x + y = 35

20 + 15 = 35

35 = 35ïƒ¼

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

You are selling tickets for a musical at your local community college.Â 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.

- WRITE A SYSTEM OF EQUATIONS:

You are selling tickets for a musical at your local community college.Â 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?

- SOLVE FOR ONE VARIABLE:
x = 175 student tickets

- SOLVE FOR THE OTHER VARIABLE:
x + y = 500

175 + y = 500

y = 325 general tickets

You are selling tickets for a musical at your local community college.Â 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?

- CHECK THE SOLUTION:(175, 325)
5(175) +8(325) = 3475

875 + 2600 = 3475

3475 = 3475ïƒ¼

x + y = 500

175 + 325 = 500

500 = 500ïƒ¼

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?

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.

- WRITE A SYSTEM OF EQUATIONS:

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?

- SOLVE FOR ONE VARIABLE:
x = 270 solid rolls

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

270 + y = 480

y = 210 printed rolls

- CHECK THE SOLUTION:(270, 210)
4(270) + 6(210) = 2340

1080 + 1260 = 2340

2340 = 2340ïƒ¼

x + y = 480

270 + 210 = 480

480 = 480ïƒ¼