What are the steps in solving word problems involving system of linear equation in two variables using the substitution or elimination method?

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.

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!!! 

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).

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.

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):

Equation (2):

Equation (3):

Putting the three equations together in a system we get:

Step 3:  Carry out the plan (solve).

*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
 

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.

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):

Equation (2):

Putting the two equations together in a system we get:

Step 3:  Carry out the plan (solve).

*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
 

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

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.  

Step 2:  Devise a plan (translate).

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:

Step 3:  Carry out the plan (solve).

*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

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

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

Step 3:  Carry out the plan (solve).

*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

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.