Transcript Solving Systems of Equations with Word Problems

Linear Systems and Problem
Solving
Ways to Solve a System of Linear Equations
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.
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.
Meg’s age is 5 times Jose’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 = Meg’s age
Let j = Jose’s age
Write an equation for each of the first two sentences.
m = 5j
5j
m + j = 18
Solve the system of equations.
How old is Meg? m  5 j
Sentence.
 5 3 
 15
(
)  j  18
6 j  18
j  3
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
110 m, find the dimensions.
let l = length
length
let w = width
width
width
length
Formula
2 l  2w 
l  2w  1
 2  18   1
2(
 36  1
 37
The width is 18 m and the
length is 37 m.
)  2 w  110
4 w  2  2 w  110
6 w  2  110
6 w  108
w  18
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
Let b = # of boys
Write an equation for each of the first two sentences.
g + b = 25
g   b
b  25
25
2

2g = 3b

 3b
 2 b  50  3 b
50  5 b
g  b  25
g  10  25
g  15
10  b
There are 15 girls and 10 boys in the class.
Example 2 The length of a rectangle is 4 m more than twice its width.
=
If the perimeter is 38
38 m, find the dimensions.
1. Labels.
let w = width
let l = length
w
l  22w
w 444
5. Sentence.
width
width
length
2. Translate first sentence. l  2w  4
3. Use perimeter formula. 2 l  2w 
4. Solve the system.
length
and
2 l  2 w  38
2(
 25   4
 10  4
 14
The width is 5 m and the length is 14 m.
)  2 w  38
4 w  8  2 w  38
6 w  8  38
6 w  30
w  5
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?
Number Labels. let a = # of adult tickets let s = # of student tickets
value of
value of
Value Labels.
let 2a =
let
1.50
s
=
adult tickets
student 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
= of each type were sold?
tickets. How many tickets
Number Labels. let a = # of adult tickets let s = # of student tickets
value of
value of
Value Labels.
let 2a =
let
1.50
s
=
adult tickets
student tickets
a = 33ss – 100
100
2a + 1.50s = 550
200

200 a  150 s  55000
  150 s
Clear the decimals.
Multiply both sides by 100.
 55000
600 s  20 , 000  150 s  55 , 000
750 s  20 , 000  55 , 000
750 s  75 , 000
s  100
a  3  100
  100
 300  100
 200
The school sold 200 adult tickets and 100 student tickets.
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. let q = # of quarters
Value Labels.
let .25q = value of quarters let .05n = value of nickels
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. let q = # of quarters
Value Labels.
let .25q = value of quarters let .05n = value of nickels
q = 3n
3n
q  32 
 6
.25q + .05n = 1.60
25 
25 q  5 n  160

Clear the decimals.
Multiply both sides by 100.
5 n  160
75 n  5 n  160
80 n  160
n  2
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.
Equations.
Let l = larger #
Let s = smaller #
5s = l + 8
s + l = 100
s   l  100
5


l 8
 5l  500  l  8
500  6l  8
492  6l
82  l
The larger number is 82 and the smaller number is 18.
Example 6 One number is 12 more than half another number. The two
numbers have a sum of 60. Find the numbers.
Assign Labels.
Equations.
Let x = first #
x 
1
2
y  12
Let y = second #
x  y  60
1

 y  12   y  60
2

1
1 y  12  60
2
 12  12
 2  3 y  48  2 
 
 
3 2
3
y  32
One number is 28 and the other number is 32.
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.
Equations.
Let m = mechanical pencils
Let p = pens
6p + m = 10 - 1
m   6 p  9
4p + 2m = 10 - 2
4p  2 
8
4 p  1 2p  18  8
 8p  1 8  8
6 1 . 25   m  9
 18  18
7 . 50  m  9
 7 . 50
 7 . 50
m  1 . 50
 8 p   10
Pens cost $1.25 each and mechanical
pencils cost $1.50 each.
8
p 
8
5
4
p  1 . 25
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?
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:





x + y = 50
2x + 5y =145
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
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?
• CHECK THE SOLUTION: (35, 15)
2(35) + 5(15) = 145
70 + 75 = 145
145 = 145 
35 + 15 = 50
50 = 50 
SOLVE THE WORD
PROBLEM:
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:





x + y = 37
2x + 3y = 80
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
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?
• CHECK THE SOLUTION: (31, 6)
2(31) + 3(6) = 80
62 + 18 = 80
80 = 80 
x + y = 37
31 + 6 = 37
37 = 37 
SOLVE THE WORD
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?
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.
• WRITE A SYSTEM OF EQUATIONS:




x + y = 35
2 x + 4y = 1 0 0
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?
• 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
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?
• CHECK THE SOLUTION:
(20, 15)
2(20) + 4(15) = 100
40 + 60 = 100
100 = 100 
x + y = 35
20 + 15 = 35
35 = 35 
SOLVE THE WORD
PROBLEM:
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?
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.
• WRITE A SYSTEM OF EQUATIONS:




x + y = 500
5x + 8y = 3475
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?
• 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. 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 = 3475 
x + y = 500
175 + 325 = 500
500 = 500 
SOLVE THE WORD
PROBLEM:
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:





x + y = 480
4x + 6y = 2340
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
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?
• CHECK THE SOLUTION: (270, 210)
4(270) + 6(210) = 2340
1080 + 1260 = 2340
2340 = 2340 
x + y = 480
270 + 210 = 480
480 = 480 