Intermediate Algebra
Tutorial 21:
Systems of Linear
Equations
and Problem Solving
 Learning Objectives
|
After completing this tutorial, you should be able to:
-
Use Polya's four step process to solve various problems involving
systems
of linear equations in both two and three variables.
|
Introduction
|
| Hey, lucky you, we have another tutorial on word
problems. As
mentioned before, whether you like it or not, whether you are going to
be a mother, father, teacher, computer programmer, scientist,
researcher,
business owner, coach, mathematician, manager, doctor,
lawyer,
banker (the list can go on and on), problem solving is
everywhere.
Some people think that you either can do it or you can't.
Contrary
to that belief, it can be a learned trade. Even the best athletes
and musicians had some coaching along the way and lots of practice -
that's
what it also takes to be good at problem solving.
The word problems in this section all involve setting up
a system of
linear equations to help solve the problem. Basically, we
are
combining the concepts from Tutorial
8: An Introduction to Problem Solving, Tutorial
19: Solving Systems of Linear Equations in Two Variables and Tutorial
20: Solving Systems of Linear Equations in Three Variables all
rolled up into one tutorial. We will be looking at different
types
of word problems involving such ideas as distance, percentages, and
something
we can all relate to MONEY!!!
|
Tutorial
|
Polya's Four Step Process
for Problem Solving
(revisited)
|
| This is the exact same process for problem solving that
was introduced
in Tutorial 8: Introduction to
Problem
Solving. The difference is in this tutorial we will be
setting
up a system of linear equations as opposed to just working with one
equation. |
| Step 1: Understand the problem. |
| Sometimes the problem lies in understanding the
problem.
If you are unclear as to what needs to be solved, then you are probably
going to get the wrong results. In order to show an understanding
of the problem you of course need to read the problem carefully.
Sounds simple enough, but some people jump the gun and try to start
solving
the problem before they have read the whole problem. Once the
problem
is read, you need to list out all the components and data that are
involved.
This is where you will be assigning your variables.
In the problems on this page,
we will be letting
each unknown be a separate variable. So, if you have
two
unknowns, you will have two variables, x
and
y.
If you have three unknowns, you will have three variables, x,
y,
and z.
|
| Step 2: Devise a plan (translate). |
| When you devise a plan (translate), you come up
with a way to
solve the problem. Setting up an equation, drawing a diagram, and
making a chart are all ways that you can go about solving your
problem.
In this tutorial, we will be setting up equations for each
problem.
You will translate them just like we did in Tutorial
2: Algebraic Expressions and Tutorial
5: Properties of Real Numbers.
In the problems on this page, we
will be setting
up systems of linear equations. The number of equations need to
match
the number of unknowns. For example, if you have two
variables,
then you will need two equations. If you have three variables,
then
you will need three equations.
|
| Step 3: Carry out the plan (solve). |
| Step 4: Look back (check
and interpret). |
| You may be familiar with the expression don't look
back. In problem
solving it is good to look back (check and interpret)..
Basically,
check to see if you used all your information and that the answer makes
sense. If your answer does check out make sure that you write
your
final answer with the correct labeling. |
 Example
1: The sum of three numbers is 14.
The
largest is 4 times the smallest, while the sum of the smallest and
twice
the largest is 18. Find the numbers. |
| Make sure that you read the question carefully several
times.
Since we are looking for three numbers, we will
let
| x = the smallest
number
y = middle number
z = the largest
number
|
|
| Since we have three unknowns,
we need to build
a system with three equations.
Equation (1):
Equation (2):
Equation (3):
|
| Putting the three equations together in a system we
get:
|
 |
*Inverse of add 4x
is sub. 4x
|
| Choose to eliminate any one of
the variables
from any pair of equations.
Since y is already
eliminated in equation
(4) and (3), it would be quickest and easiest to eliminate y.
We can use equation (4) as one equation with y
eliminated:
|
 |
*y
is already
eliminated |
Eliminate the SAME variable
chosen in step
2 from any other pair of equations, creating a system of two equations
and 2 unknowns.
We can use equation (3) as another equation with y
eliminated:
|
 |
*y
is already
eliminated |
| Solve the remaining system
found in step 2
and 3.
Putting those two equations together we get:
|
| I’m going to choose to eliminate z.
Multiplying equation (4) by -2 and then adding that
to equation (3)
we get:
|
 |
*Mult. both sides of eq. (4)
by -2
*z's
have opposite
coefficients
*z's
dropped out
|
 |
*Inverse of mult. by 9 is div. by 9 |
| Using equation (4) to plug 2 in for
x
and solving for z we get: |
 |
*Equation (4)
*Plug in 2 for x
*Inverse of sub. 8 is add 8
|
| Solve for the third variable.
Using equation (1) to plug in 2 for
x
and 8 for z and solving for y
we get:
|
 |
*Equation (1)
*Plug in 2 for
x and 8 for z
*Inverse of add 10 is sub. 10
|
You will find that if you plug the ordered triple (2,
4, 8) into ALL
THREE equations of the original system, this is a solution to ALL
THREE of them.
Final Answer:
2 is the smallest number, 4 is the middle number and 8
is the largest
number.
|
| Example
2: How many gallons of 20% alcohol
solution
and 50% alcohol solution must be mixed to get 9 gallons of 30% alcohol
solution? |
| Make sure that you read the question carefully several
times.
Since we are looking for two different amounts, we will
let
| x = the number
of gallons of 20% alcohol
solution
y = the number of
gallons of 50% alcohol
solution
|
|
| Since we have two unknowns, we
need to build
a system with two equations.
Equation (1):
Equation (2):
|
| Putting the two equations together in a system we
get:
|
 |
*Mult. both sides of eq. (2) by 10
|
 |
*Mult. both side of eq. (1) by
-2
*x's
have opposite
coefficients
*x's
dropped out
|
| Solve for remaining variable.
Solving for y we get:
|
 |
*Inverse of mult. by 3 is divide
by 3 |
| Solve for second variable.
Using equation (1) to plug in 3 for y
and solving for x we get:
|
 |
*Equation (1)
*Plug in 3 for y
*Inverse of add 3 is sub. 3
|
You will find that if you plug the ordered pair (6, 3)
into BOTH equations
of the original system, that this is a solution to BOTH of them.
Final Answer:
6 gallons of 20% solution and 3 gallons of 50% solution
|
| Example
3: An airplane flying with the wind can
cover
a certain distance in 2 hours. The return trip against the wind
takes
2.5 hours. How fast is the plane and what is the speed of the
air,
if the one-way distance is 600 miles? |
| Make sure that you read the question carefully several
times.
Since we are looking for two different rates, we will
let
| x = rate of the
plane
y = the rate of
the wind
|
Since this is a rate/distance problem, it might be good
to organize
our information using the distance formula.
Keep in mind that the wind speed is affecting the
overall speed.
When the plane is with the wind, it will be going
faster. That
rate will be x + y.
When the plane is going against the wind, it will be
going slower.
That rate will be x -
y.
|
|
(Rate)
|
(Time)
|
= Distance
|
|
With wind
|
x + y
|
2
|
600
|
|
Against wind
|
x - y
|
2.5
|
600
|
|
| Since we have two unknowns, we
need to build
a system with two equations.
Equation (1):
Equation (2):
|
| Putting the two equations together in a system we
get:
|
 |
*Div. both side of eq. (1) by 2
*Div. both sides of eq. (2) by
2.5
|
 |
*y's
have opposite
coefficients
*y's
dropped out
|
| Solve for remaining variable.
Solving for x we get:
|
 |
*Inverse of div. by 2 is mult.
by 2
|
| Solve for second variable.
Using equation (3) to plug in 270 for x and solving
for y we get:
|
 |
*Equation (3)
*Plug in 270 for x
*Inverse of add 270 is sub. 270 |
| You will find that if you plug the ordered pair (270,
30) into BOTH
equations of the original system, this is a solution to BOTH of
them.
Final Answer:
The airplane speed is 270 mph and the air speed is 30 mph
|
| Example
4: Given the cost function C(x)
and the revenue function R(x), find the
number
of units x that must be sold to break
even.
C(x) = 20x
+
50000
R(x) = 25x
|
| Make sure that you read the question carefully several
times.
We will let,
| x = the number
of units
C(x) = 20x
+ 50000
R(x) = 25x
|
This problem appears a little different because of the
function notation.
Keep in mind that function notation translates to being y.
|
| In this problem, the two
equations that we are
working with have already been given to us:
Cost function:
C(x) = 20x
+ 50000
Revenue function:
R(x) = 25x
|
 |
*Inverse of add 20x
is sub. 20x
*Inverse of mult. by 5 is div.
by 5
|
You will find that if you plug 10000 for x
into BOTH equations that they BOTH come out to be 250000, which means
this
is the break-even point.
Final Answer:
10000 units are needed to break-even
|
Practice Problems
|
| These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer.
|
 Practice
Problems 1a - 1c:
Solve.
|
1a. The larger of two numbers is 5 more than
twice the smaller.
If the smaller is subtracted from the larger, the result is 12.
Find
the numbers.
(answer/discussion
to 1a) |
1b. It takes a boat 2 hours to travel 24 miles
downstream and
3 hours to travel 18 miles upstream. What is the speed of the
boat
in still water and of the current of the river?
(answer/discussion
to 1b) |
1c. A student has money in three accounts that
pay 5%, 7%, and
8%, in annual interest. She has three times as much invested at
8%
as she does at 5%. If the total amount she has invested is $1600
and her interest for the year comes to $115, how much money does she
have
in each account?
(answer/discussion
to 1c) |
Need Extra Help on These Topics?
|
All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on Jan. 7, 2002 by Kim Seward. |
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to the THEA Math Help Page)
