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

Learning 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.

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 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!!!

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

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):

Equation (2):

Equation (3):

Putting the three equations together in a system we get:

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:

*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 Solving for x we get:

*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.

Mixing Solutions

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):

Equation (2):

Putting the two equations together in a system we get:

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:

*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: *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

Distance/Rate

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.

With wind Against wind

(Rate)

(Time)

= Distance

x+y x-y

2

600

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):

Equation (2):

Putting the two equations together in a system we get:

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:

*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:

*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

Break-even Point

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.

Step 3: Carry out the plan (solve). Solving for x we get:

*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

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?

There were no good websites found to help us with the topics on this page. Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

(Back to the College Algebra Homepage) All contents copyright (C) 2002, WTAMU and Kim Peppard. All rights reserved. Last revised on November 13, 2002 by Kim Peppard.