Learning Objectives

1. Use Polya's four step process to solve various problems involvingsystemsof linear equations in both two and three variables.

Introduction

The word problems in this section all involve setting upa system oflinear equations to help solve the problem.  Basically,  wearecombining the concepts from allrolled up into one tutorial.  We will be looking at differenttypesof word problems involving such ideas as distance, percentages, andsomethingwe can all relate to MONEY!!!  

Tutorial

In the problems on this page,we will be lettingeach unknown be a separate variable.  So, if you havetwounknowns, you will have two variables, xand y. If you have three unknowns, you will have three variables, x, y,and z.

In the problems on this page, wewill be settingup systems of linear equations.  The number of equations need tomatchthe number of unknowns.  For example, if you have twovariables,then you will need two equations.  If you have three variables,thenyou will need three equations.

Since we are looking for three numbers, we willlet 
 
 

Equation (1):

Equation (2):

Equation (3):

Simplify and put all threeequations in theform Ax + By  + Cz = D if needed.

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

Since y is alreadyeliminated in equation(4) and (3), it would be quickest and easiest to eliminate y.

We can use equation (4) as one equation with yeliminated:

We can use equation (3) as another equation with yeliminated:

Putting those two equations together we get:

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

*z'shave oppositecoefficients

*z'sdropped out

*Inverse of sub. 8 is add 8
 

Using equation (1) to plug in 2 forxand 8 for z and solving for ywe get:

*Inverse of add 10 is sub. 10
 

Final Answer:
2 is the smallest number, 4 is the middle number and 8is the largestnumber.

Since we are looking for two different amounts, we willlet 
 
 

Equation (1):

Equation (2):

Simplify if needed. 

We can simplify this by multiplying both sides ofequation (2) by10 and getting rid of the decimals:

Multiply one or both equationsby a numberthat will create opposite coefficients for either x or y if needed ANDadd the equations.

If we multiply equation (1) by -2,then the x’swill have opposite coefficients. 

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

*x'shave oppositecoefficients
 

*x'sdropped out

Solving for y we get:

Using equation (1) to plug in 3 for yand solving for x we get:

*Inverse of add 3 is sub. 3
 

Final Answer:
6 gallons of 20% solution and 3 gallons of 50% solution

Since we are looking for two different rates, we willlet 
 
 

Since this is a rate/distance problem, it might be goodto organizeour information using the distance formula.

Keep in mind that the wind speed is affecting theoverall speed. 

When the plane is with the wind, it will be goingfaster.  Thatrate will be x + y. 

When the plane is going against the wind, it will begoing slower. That rate will be x -y.
 
 

Equation (1):

Equation (2):

Simplify if needed. 

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

*Div. both sides of eq. (2) by2.5
 
 
 

Multiply one or both equationsby a numberthat will create opposite coefficients for either x or y if needed ANDadd the equations.

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

*y'sdropped out

Solving for x we get:

Using equation (3) to plug in 270 for x and solvingfor y we get:

Final Answer:
The airplane speed is 270 mph and the air speed is 30 mph

C(x) = 20x+50000
R(x) = 25x

We will let,
 

This problem appears a little different because of thefunction notation.Keep in mind that function notation translates to being y.

Cost function:
C(x) = 20x+ 50000

Revenue function:
R(x) = 25x

Since the break-even point iswhen revenue= cost, we will go right into setting this up using the substitutionmethod as discussed in Tutorial 19: Solving Systems of Linear Equationsin Two Variables.

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

Final Answer:
10000 units are needed to break-even

Practice Problems

To get the most out of these, you should work theproblem out onyour own and then check your answer by clicking on the link for theanswer/discussionfor that  problem.  At the link you will find the answeras well as any steps that went into finding that answer.

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