### Solving Applications: Systems of Two Equations

```Section 3.3
Applications: Systems of Equations

Sum & Difference Problems F + S = T
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
Total-Value Problems Num · UC = TV


Principal x Rate x Time = Interest
Purity Problems Num · Pct = Pure


Number x Unit Price = Total Value
Interest Problems Prin · Rate · T = Int


First + Second = Total Amount
Number x Percent = Pure Amount
Motion Problems R · T = D

Rate x Time = Distance
3.3
1
Application Problem H/W Format
1. Jewelry Design. To make a necklace, a jewelry designer bought 80 beads for a
total of \$39. Some were silver beads (40 cents each) and the rest were gemstone
1. Jewelry
Design.
Making a(can
\$39 necklace
…
#.
Title First
sentence
be abbreviated)
withand
____
and
names
Table· Formula
Num
UC = TV
____ (s)silver
____
(g)gemstone
Silvers
A Equation
Gemstones
B Equation
total
work work work
work work work
to find the numbers to put into3.3the answer stmt
2
Sum & Difference Example p155, #42
In 2008, there were 146 (t)hreatened plant species and 600 (e)ndangered plant species
42.
The sum of two numbers is 30. The first
number is twice the second number. What are
the numbers?
42. The sum of two numbers is 30.
The numbers are 20 (x) and 10 (y)
A x  y  30
B x  2y
B  A 2 y  y  30
3 y  30
y  10
B x  2(10)  20
chk A 20  10  30
3.3
3
Sum & Difference Example #2
1. Endangered Species. The number of plant species listed as
threatened or endangered has more than tripled in the last 20 years. In
2008 there were 746 plant species listed as either threatened or
endangered. The number threatened was 4 less than ¼ the number
endangered. How many plant species were endangered and how
many were threatened?
1. Endangered Species. …tripled in last 20 years.
A t  e  746
Threatened: 146 (t), Endangered 600 (e)
B t  14 e  4
A t  e  746
4 B 4t  e  16
5t  730, so t  146
A e  746 t  746 146  600
chk B 146  1 4 (600)  4  150 4
3.3
4
(Your textbook does not have this overview)
Setting up Tables to Solve Problems



Determine the appropriate formula
Num · UC = TV
Draw a table:






Make the columns match the formula
Make one row for each item
(or each situation), and label it
Add a Total or Mixture row, if appropriate
Use two meaningful letters for the unknowns (see answer stmt)
Fill in two of the columns with letters and numbers, and compute
the expressions for the other column or for the Total row
Find 2 different equations from the table relationships
3.3
5
Total Value Example
1. Jewelry Design. To make a necklace, a jewelry designer bought 80
beads for a total of \$39. Some were silver beads (40 cents each) and
the rest were gemstone beads (65 cents each). How many beads of
each type did the designer buy?
1. Jewelry Design. Make a necklace of 80 beads …
Bought: 52
3.3
6
Total Value Example #2
2.sPurchasing. Recently the Woods County Art Center
purchased 120 stamps for \$33.90. If the stamps were a
combination of 23¢ postcard stamps and 37 ¢ first-class stamps,
how many of each type were bought?
2. Purchasing. Woods County Art Center stamps…
__ (p)ostcard and 75
__ 1st-class(f) stamps
They bought 45
3.3
7
Total Value Example #3
Blending Teas. Sonya’s House of Tea sells loose Lapsang Souchong tea for 95¢ an
ounce and Assam Gingia for \$1.43 an ounce. Sonya wants to make a 20-oz mixture
of the two types, called Dragon Blend, that will sell for \$1.10 an ounce. How much
tea of each type should Sonya use?
3. Blending teas. Sonya’s House …Dragon’s Blend
Use 13.75
___ oz. of (A)ssam
___ oz. of (L)apsang 6.25
3.3
8
Interest Example
6. Student Loans. Ranjay’s student loans totaled \$9600. Part was a
Perkins loan made at 5% interest and the rest was a Stafford loan
made at 8% interest. After one year, Ranjay’s loans accumulated
\$633 in interest. What was the original amount of each loan?
6. Student Loans. Total was \$9600, interest was \$633
(p)Perkins loan = \$4500
____
____and (s)Stafford loan was = \$5100
3.3
9
Purity Example
4. Mixing Fertilizers. Sky Meadow Gardening, Inc., carries two brands of liquid
fertilizer containing nitrogen and water. “Gently Green” is 5% nitrogen and
“Sun Saver” is 15% nitrogen. The company needs to combine the two types of
solutions to make 90 L of a solution that is 12% nitrogen. How much of each
brand should be used?
4. Mixing Fertilizers. Creating a 12% nitrogen fertilizer…
__ L of (S)S
__ L of (G)G with 63
The Co. should mix 27
3.3
10
Motion (same direction) Example
7. Train Travel. A Vermont Railways freight train, loaded with logs, leaves
Boston, bound for Washington D.C. at a speed of 60 km/h. Two hours later,
and Amtrak Metroliner leaves Boston bound for Washington D.C., on a parallel
track at 90 km/h. At what point will the Metroliner catch up to the freight
train?
7. Train Travel. Boston to Washington, two trains…
The Amtrak will catch up with the Freight 360
___km from Boston
3.3
11
Motion (winds & currents) Example
8. Jet Travel. A Boeing 747-400 jet flies 4 hours west with a 60 mph tailwind.
Returning against the wind takes 5 hours. Find the speed of the plane with no
wind.
The speed of the plane with no wind is 540
___mph
3.3
12
What Next?

Section 3.4 –
Systems of Three
Equations
3.3
13
```