Transcript Algebra 2

Algebra 2
2.3: Applications of Linear Equations
(AKA: WORD PROBLEMS)
Verbal Expressions
Addition
Verbal Expressions
Subtraction
Verbal Expressions
Multiplication
Verbal Expressions
Division
Sentences to Equations
Problems
Determine if the following are expressions or equations:
a) 2(3  x)  4 x  7
b) 2(3  x)  4 x  7  1
Expression
Equation
(No equals sign)
(Has an equals sign)
Solving an Applied Problem
1. Read the problem (several times) until you
understand what is given and what needs to
be found.
2. Assign a variable. Use diagrams or tables to
help you out!!!
3. Write an equation. The equation can have
more than one variable but it must be the
same variable!!!
4. Solve the equation.
5. State the answer and check if needed.
Problems
The length of a rectangle is 1 cm more than twice the
width. The perimeter of the rectangle is 110 cm.
Find the length and the width of the rectangle.
P  2L  2W
110  2(2 x  1)  2( x)
length  2 x  1
110  4x  2  2x
110  6x  2
2
2
width  18cm
108  6 x
6
6
length  2(18)  1  37cm
18  x
width  x
Your turn!!!
The length of a rectangle is 5 cm more than
three times the width. The perimeter of the
rectangle is 350 in. Find the length and the
width of the rectangle. P  2L  2W
350  2(3x  5)  2( x)
width  x
350  6x  10  2x
length  3x  5
350  8x  10
 10
 10
width  42.5in.
340  8x
8
8
length  3(42.5)  5  132.5in.
42.5  x
Problems
Two outstanding major league pitchers in recent years are Randy Johnson
and Pedro Martinez. In 2002, they combined for a total of 573
strikeouts. Johnson had 95 more strikeouts than Martinez. How many
strikeouts did each pitcher have?
J  M  573
( x  95)  ( x)  573
J  x  95
2x  95  573
 95  95
M  239 strikeouts
2 x  478
J  239  95
2
2
 334 strikeouts
x  239
M x
Your turn!!!
At the end of the 2003 baseball season, Sammy
Sosa and Barry Bonds had a lifetime total of
1197 home runs. Bonds had 119 more than
Sosa. How many home runs did each player
have?
B  S  1197
Sx
( x  119)  ( x)  1197
B  x  119
2x  119  1197
S  539 Homeruns
 119  119
J  539  119
2 x  1078
x  539
 658 Homeruns
2
2
Problems
After winning the state lottery, Mark LeBeau has $40,000 to invest. He
will put part of the money in an account that earns 4% interest and the
remainder in stocks paying 6% interest. His accountant told him that
this should result in $2040 in interest per year. How much should be
invested at each rate?
Rate
Principal
Interest
.04
X
.04x
.06
40,000-x
.06(40,000-x)
40,000
2040
.04  $18,000
.06  $40,000 - $18,000
 $22,000
.04( x)  .06(40,000  x)  2040
.04x  2400  .06x  2040
 .02x  2400  2040
 2400  2400
 .02x  360
 .02  .02
x  18,000
Your turn!!!
A man has $34,000 to invest. He invests some
at 5% and the balance at 4%. His total
annual interest income is $1,545. Find the
amount he invests at each rate.
Rate
Principal
Interest
.05
X
.05x
.04
34,000-x
.04(34,000-x)
34,000
1545
.05  $18,500
.04  $34,000 - $18,500
 $15,500
.05( x)  .04(34,000  x)  1545
.05x  1360  .04x  1545
.01x  1360  1545
1360  1360
.01x  185
.01 .01
x  18,500
Problems
A chemist must mix 8 L of a 40% acid solution with some 70% solution to
get it to a 50% acid solution. How much of the 70% solution should be
used?
%
Amount
Total
40%
8L
.4(8)
70%
xL
.7(x)
50%
(8 + x) L
.5(8 + x)
Equation
.4(8)  .7( x)  .5(8  x)
3.2  .7 x  4  .5x
 .5x
 .5x
3.2  .2 x  4
 3.2
 3.2
.2 x  .8
.2 .2
x  4 Liters
Your turn!!!
How many liters of a 10% solution should be
mixed with 60 L of a 25% solution to get a
15% solution?
.10( x)  .25(60)  .15( x  60)
Amount Total
%
.10x  15  .15x  9
10%
x
.10x
 15
 15
25%
60
.25(60)
15%
x+60
.15(x+60)
.10x  .15x  6
 .15x  .15x
 .05x  6
 .05 - .05
x  120 Liters
Problems
In 2002, there were 301 area codes in the United States. This was a
250% increase from 1947 when area codes were first established.
How many area codes were the in 1947?
Total  301
original (1947)  increase (2001)  301
2001  2.5 x
x  2.5x  301
(250% of x)
3.5x  301
3.5 3.5
x  86
1947  x
1947  x  86 area codes
Problems
The octane rating of gasoline is a measure of its antiknock qualities. For a
standard fuel, the octane rating is the percent of isooctane. How many
liters of pure isooctane should be mixed with 200 L of 94% isooctane to
get a mixture that is 98% isooctane?
%
Amount
Total
100%
xL
1(x)
94%
200 L
.94(200)
98%
(x + 200)
.98(x + 200)
Equation
1( x)  .94(200)  .98( x  200)
x  188  .98x  196
 .98x
 .98x
.02 x  188  196
 188  188
.02 x  8
.02 .02
x  400 Liters