College Algebra Tutorial 51


College Algebra
Tutorial 51: Systems of Linear Equations
and Problem Solving


WTAMU > Virtual Math Lab > College Algebra

 

deskLearning Objectives



After completing this tutorial, you should be able to:
  1. Use Polya's four step process to solve various problems involving systems of linear equations in both two and three variables.




deskIntroduction



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 16: Formulas and Applications, Tutorial 49: Solving Systems of Linear Equations in Two Variables and Tutorial 50: 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!!! 

 

 

desk Tutorial


 
 
  Polya's Four Step Process 
for Problem Solving 
(revisited)
 

This is the exact same process for problem solving that was introduced in Tutorial 16: Formulas and Applications  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. 

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).

 
The next step, carry out the plan (solve), is big. This is where you solve the system of equations you came up with in your devise a plan step.  The equations in the systems in this tutorial will all be linear equations.  If you need help solving them, by all means, go back to Tutorial 49: Solving Systems of Linear Equations in Two Variables and Tutorial 50: Solving Systems of Linear Equations in Three Variables  and review the concepts.

 
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.

 
 
  Numeric Word Problem
 
notebook 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.

 
Step 1:  Understand the problem.

 
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
 
 

Step 2:  Devise a plan (translate).

 
Since we have three unknowns, we need to build a system with three equations.

Equation (1):

example 1a

Equation (2):

example 1b

Equation (3):

example 1c
 

Putting the three equations together in a system we get:

example 1d
 
 

Step 3:  Carry out the plan (solve).

 
This is a system of linear equations with three variables, which can be found in Tutorial 50: Solving Systems of Linear Equations in Three Variables
 

Simplify and put all three equations in the form Ax + By  + Cz = D if needed.

Equation (2) needs to be put in the correct form:
 

example 1e

*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:
 

example 1f
*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:
 

example 1g
*y is already eliminated

 
Solve the remaining system found in step 2 and 3.

Putting those two equations together we get:
 

example 1h

 
I'm going to choose to eliminate z.

Multiplying equation (4) by -2 and then adding that to equation (3) we get:
 

example 1i
*Mult. both sides of eq. (4) by -2
 
 
 

*z's have opposite coefficients

*z's dropped out
 

Solving for x we get:

 
example 1j

*Inverse of mult. by 9 is div. by 9

 
Using equation (4) to plug 2 in for x and solving for z we get:

 
example 1k
*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:
 

example 1l
*Equation (1)
*Plug in 2 for x and 8 for z
 

*Inverse of add 10 is sub. 10
 
 
 

Step 4:  Look back (check and interpret).

 
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.
 
 
  Mixing Solutions
 

notebook Example 2:    How many gallons of 20% alcohol solution and 50% alcohol solution must be mixed to get 9 gallons of 30% alcohol solution?

 
Step 1:  Understand the problem.

 
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

 
 

Step 2:  Devise a plan (translate).

 
Since we have two unknowns, we need to build a system with two equations.

Equation (1):

example 2a

Equation (2):

example 2b
 

Putting the two equations together in a system we get:

example 2c
 
 

Step 3:  Carry out the plan (solve).

 
This is a system of linear equations with two variables, which can be found in Tutorial 49: Solving Systems of Linear Equations in Two Variables.

Simplify if needed. 

We can simplify this by multiplying both sides of equation (2) by 10 and getting rid of the decimals:
 

example 2d

*Mult. both sides of eq. (2) by 10
 

 
 

At this point, you can use any method that you want to solve this system.  I'm going to use the elimination method as discussed in Tutorial 49: Solving Systems of Linear Equations in Two Variables.
 

Multiply one or both equations by a number that will create opposite coefficients for either x or y if needed AND add the equations.

If we multiply equation (1) by -2, then the x's will have opposite coefficients. 

Multiplying equations (1) by -2 and then adding that to equation (3) we get:
 

example 2e
*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:
 

example 2f

*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:
 

example 2g
*Equation (1)
*Plug in 3 for y

*Inverse of add 3 is sub. 3
 
 
 

Step 4:  Look back (check and interpret).

 
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
 
 
  Distance/Rate
 

notebook 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?

 
Step 1:  Understand the problem.

 
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
 
 

Step 2:  Devise a plan (translate).

 
Since we have two unknowns, we need to build a system with two equations.

Equation (1):

example 3a

Equation (2):

example 3b
 

Putting the two equations together in a system we get:

example 3c
 
 

Step 3:  Carry out the plan (solve).

 
This is a system of linear equations with two variables, which can be found in Tutorial 49: Solving Systems of Linear Equations in Two Variables.

Simplify if needed. 

We can simplify this by dividing both sides of equation (1) by 2 and equation (2) by 2.5 getting rid of the ( ) and decimals at the same time:
 

example 3d
*Div. both side of eq. (1) by 2

*Div. both sides of eq. (2) by 2.5
 
 
 

 
 

At this point, you can use any method that you want to solve this system.  I'm going to use the elimination method as discussed in Tutorial 49: Solving Systems of Linear Equations in Two Variables.
 

Multiply one or both equations by a number that will create opposite coefficients for either x or y if needed AND add the equations.

Since we already have opposite coefficients on y, we can go right into adding equations (3) and (4) together:
 

example 3e

*y's have opposite coefficients

*y's dropped out
 

Solve for remaining variable.

Solving for x we get:
 

example 3f
*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:
 

example 3g
*Equation (3)
*Plug in 270 for x
*Inverse of add 270 is sub. 270

 
 
Step 4:  Look back (check and interpret).

 
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
 
 
 
  Break-even Point
 

notebook 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
 

Step 1:  Understand the problem.

 
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.
 
 

Step 2:  Devise a plan (translate).

 
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
 

Putting the two equations together in a system we get:

Since the break-even point is when revenue = cost, we will go right into setting this up using the substitution method as discussed in Tutorial 49: Solving Systems of Linear Equations in Two Variables.

example 4b
 
 

Step 3:  Carry out the plan (solve).

 
Solving for x we get:

 
example 4a
*Inverse of add 20x is sub. 20x
 

*Inverse of mult. by 5 is div. by 5

 
 
 

Step 4:  Look back (check and interpret).

 
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
 

 
 

desk 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.

pencil Practice Problems 1a - 1c: Solve each word problem using systems of equations.


 

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)

 

 

 

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WTAMU > Virtual Math Lab > College Algebra


Last revised on April 29, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.