Name __________________________________
Period __________
Date:
3-6
Problem
Solving Using
Systems
Essential Question Suppose you run a business that makes
two products. Each product requires a different amount of
a raw material and sells for a different price, but you have
a fixed amount of the raw material, and you want to
receive a precise amount for the sale of your entire
inventory. How could you determine the amount of each
product to produce in order to use all your raw material
and achieve the desired amount for the sale of your
products?
Standard: A-REI.6
Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
Objective:
To use systems of equations to solve problems.
Topic:
Sometimes you can solve a problem involving two unknown
quantities most easily by using a system of equations. The
five-step problem solving plan discussed in an earlier lesson is
useful here. However, you must set up and solve a system of
equations rather than a single equation.
In this lesson, you will learn how to solve problems using a
system of equations by studying examples. Be sure you
carefully explain the examples and completely work all the
exercises.
Example 1:
Summary
To use a certain computer data base, the charge is $30/h during
the day and $10.50/h at night. If a research company paid $411
for 28 h of use, find the number of hours charged at the
daytime rate and at the nighttime rate.
Step 1 The problem asks for the number of hours charged at each rate.
Step 2 Let d = the number of hours of use at the daytime rate.
Let n = the number of hours of use at the nighttime rate.
Step 3 Set up a system of equations.
Total number of hours charged is 28.
Total amount charged is 411.
Step 4 Solve the system.
Express d in terms of n in the first equation and
substitute in the second equation.
Since
(6, 22).
, and
, the solution is
Step 5 Check



there were 6 h charged at the daytime rate and 22 h at the
nighttime rate.
2
Kelly asked a bank teller to cash a $390 check using $20 bills
and $50 bills. If the teller gave her a total of 15 bills, how
many of each type of bill did she receive?
Exercise 1:
Step 1
Step 2
Step 3
Step 4
Step 5
3
Vocabulary:
The following terms are used in connection with aircraft flight.
air speed the speed of an aircraft in still air
wind speed the speed of the wind
tail wind a wind blowing in the same direction as the path of the aircraft
head wind a wind blowing in the direction opposite to the path of the
aircraft
ground speed the speed of the aircraft relative to the ground
This vocabulary is commonly used in problems involving
flight. The following example illustrates this.
Example 2:
To measure the speed of the jet stream (a high-speed, highaltitude, west-to-east wind), a weather-service plane flew 1800
km with the jet stream as a tail wind and then back again. The
eastbound flight took 2 h, and the westbound return flight took
3 h 20 min. Find the speed of the jet stream and the air speed
of the plane.
Step 1 The problem asks for the plane's air speed and the speed of the
wind.
Step 2 Let p = the air speed in km/h.
Let w = the wind speed in km/h.
Using the fact that rate  time = distance, construct a table for
the plane’s ground speed.
rate
Eastbound

time
=
distance
2
Westbound
4
Step 3 From the right-hand column of the table, the plane flew a total
distance east of
and a total distance west of
These distances are the same, and they are equal to 1800 km.
Therefore, we have the following two equations:
Step 4 Solve the system.
Using the linear-combination method,
Step 5 Check



the air speed of the plane is 720 km/h and the speed of
the jet stream is 180 km/h.
5
With a tail wind, a helicopter flew 300 mi in 1 h 40 min. The return
trip against the same wind took 20 min longer. Find the wind speed
and the air speed of the helicopter.
Exercise 2:
Step 1
Step 2
Using the fact that rate  time = distance, construct a table for the
helicopter’s ground speed.
rate

time
=
distance
Outbound
Return
Step 3
6
Step 4
Step 5
7
Example 3:
Davis Rent-A-Car charges a fixed amount per weekly rental
plus a charge for each mile driven. A one-week trip of 520
miles cost $250, and a two-week trip of 800 miles cost $440.
Find the weekly charge and the charge for each mile driven.
Step 1 The problem asks for the weekly charge and the charge for
each mile driven.
Step 2 Let
Let
.
.
Therefore, the cost of renting a car is,
,
where x is the number of weeks, and y is the number of miles.
Step 3 For the one-week trip of 520 miles
.
For the two-week trip of 800 miles
.
Step 4 Solve the following system of equations:
.
.
Multiply the first equation by 2 and add it to the second
equation.
8
Substituting
into the first, original equation,
,
or
.
Step 5 Davis Rent-A-Car charges $120 per week plus $0.25 per mile
driven.
Check


A deep sea charter boat company charges fishermen an hourly
fee plus a charge for each large fish caught. A six-hour fishing
trip resulting in 7 large fish caught cost the fishermen $925,
and a four-hour fishing trip resulting in 19 large fish caught
cost the fishermen $975. What is the hourly charge for
chartering the fishing boat, and what is the charge for each
large fish caught?
Exercise 3:
Step 1
Step 2
Step 3
9
Step 4
Step 5
10
Example 4:
If a particle starting with an initial speed of v0 has a constant
acceleration a, then its speed after t seconds is given by
.
Find v0 and a if
when
, and
when
.
Step 1 The problem asks for the values of the initial speed and the
constant acceleration.
Step 2 Let v0 = the initial speed.
Let a = the constant acceleration.
Step 3 Then,
and
.
Step 4 We’ll solve this system using the substitution method.
Solving the first equation for v0 gives,
Substituting this expression for v0 into the second equation
gives,
.
Solving for a gives,
.
Substituting for a in the equation for v0 gives,
11
Step 5
⁄
and
⁄.
Check


You may have seen a video of an astronaut dropping a hammer
and a feather in the airless environment of the moon. This was
intended to demonstrate that objects fall with the same
acceleration in the absence of air-resistance. You may have
also noticed that the hammer fell much more slowly than it
would have if it were dropped on the earth. This is because the
moon has a smaller gravity than the earth does.
Exercise 4:
Suppose a satellite orbiting the moon in a close orbit (close to
the lunar surface) ejected a probe that struck the moon’s
surface. This probe was ejected in such a way that it had no
horizontal velocity. At the instant the probe left the satellite
was traveling directly down with a velocity of v0. If the
moon’s gravitational acceleration is gm, then the probe’s
downward speed is given by the following equation:
where t is the time after the probe’s release. Moreover, 15 s
after its release, the probe was observed to be traveling at
47 m/s and 90 s after its release, the probe was traveling at
167 m/s
What is the moon’s gravitational constant (gm), and what was
the downward speed of the probe when it was released?
Step 1
12
Step 2
Step 3
Step 4
Step 5
Check
Class work:
none
Homework:
p 132 Problems: 2-10 even
P 133 Problems: 11-14, 16
13