Solving Systems of Equations in Three Variables

Solving Systems of Equations in Three Variables

Section 2-2Section 2-2

Before finishing this section you should be able to:

Solve systems of equations involving

three variables algebraically

Remember: Your textbook is your friend! This presentation is just a supplement to the text. BEFORE you view this, make sure you read this section in your textbook and look at all the great examples that are also worked there for you.

We can solve systems of equations with three variables algebraically with the same methods we

used to solve for two variables. The solution is an ordered triple (x, y, z).

Solve the system of equations by elimination.x + 4y – z = 203x + 2y + z = 8

2x – 3y + 2z = -16

One way to solve a system of three equations is to choose pairs of equations and then eliminate one of the variables. Because the coefficient of x is 1 in the first equation, it is a good choice for

eliminating x from the second and third equations.

To eliminate x using the first and second equations, multiply each side of the first equation by –3.

-3(x + 4y – z) = -3(20)-3x – 12y + 3z = -60

Then add that result to the second equation.-3x – 12y + 3z = -60

3x + 2y + z = 8-10y+4z =-52

To eliminate x using the first and third equations, multiply each side of the first equation by –2.

-2(x + 4y – z) = -2(20)-2x – 8y + 2z = -40

Then add that result to the third equation.-2x– 8y +2z = -402x – 3y +2z = -16

-11y+4z = -56

Now you have two linear equations in two variables. Solve this system. Eliminate z by multiplying each side of the second equation by –1 and adding the two equations.

-10y + 4z = -52-1(-11y + 4z) = -1(-56)

-10y + 4z = -5211y – 4z =56 y = 4

The value of y is 4.

By substituting the value of y into one of the equations in two variables, we can solve for the value of z.

-10y + 4z = -52-10(4) + 4z = -52 y = 4

4z = -12

z = -3 The value of z is –3.

Finally, use one of the original equations to find the value of x.

x + 4y – z = 20x + 4(4) – (-3) = 20 y = 4, z = -3

x = 1

The solution is x = 1, y = 4, and z = -3. This can be written as the ordered triple (1, 4, -3)

Check by substituting the values into each of the original equations.

Solve the system of equations by substitution.2x = -6y

x + y + z = 10-4x – 4y – z = -4

You can easily solve the first equation for x.2x = -6y

x = -3y Divide each side by 2.

Then substitute –3y for x in each of the other two equations. Simplify each equation…

x + y + z = 10-3y + y + z = 10 x = -3y

-2y + z = 10

-4x – 4y – z = -4-4(-3y) – 4y – z = -4 x = -3y

8y – z = -4

Solve –2y + z = 10 for z. -2y + z = 10

z = 10 + 2y Add 2y to each side.

Substitute 10 + 2y for z in 8y – z = -4. Simplify.

8y – (10 + 2y) = -4 z = 10 + 2y6y – 10= -4

y = 1

Now, find the values of z and x. Use z = 10 + 2y and x = -3y. Replace y with 1.

z = 10 + 2y x = -3y z = 10 + 2(1) y = 1 x = -3(1)

z = 12 x = -3

The solution is x = -3, y = 1, and z = 12 (-3, 1, 12)Check each value in the original system.

MANUFACTURING A manufacturer of golf balls supplies three driving ranges with balls. The output from a week’s production of balls is 320 cases. The manufacturer must send driving range A three times as many cases as are sent to driving range B, and must send driving range C 160 cases less than ranges A and B together. How many cases should be sent to each driving range to distribute the entire week’s production to them?

Write a system of equations. Define the variables as follows.x = the number of cases sent to driving range Ay = the number of cases sent to driving range Bz = the number of cases sent to driving range C

Real-World Example

The system is:x + y + z = 320 total number of cases produced x = 3y cases to A equal three times cases to B

z = x + y – 160 cases to C equal 160 less than A and B togetherThe second equation tells us that 3y can be substituted

for x in the other two equations. Simplify each equation.

x + y + z = 320 3y + y + z = 320 x = 3y

4y + z = 320

z = x + y – 160z = 3y + y – 160 x = 3y

z = 4y - 160

Substitute 4y – 160 for z in 4y + z = 320.

4y + z = 320

4y + 4y – 160 = 320 z = 4y - 1608y – 160 = 320

y = 60Now, find the values of x and z. Use x = 3y and z = 4y – 160.

Replace y with 60.

x = 3y z = 4y – 160x = 3(60) y = 60 z = 4(60) – 160 y = 60

x = 180 z = 80

The manufacturer should send 180 cases of golf balls to driving range A, 60 cases to driving range B, and

80 cases to driving range C.

First we must eliminate one of the variables from two of the equations. Let’s get rid of the y-variable. (If any variable in any of the three equations has a coefficient of 1that is going to be the easiest variable to eliminate)To eliminate y using the first and second equations, multiply each side of the second equation by 2.Then add that result to the second equation.Now we have a linear equation in two variables.

5x-2y+z = 112x+y+3z = 06x-2y-2z = 16 5x-2y+z = 112(2x+y+3z) = 2(0)5x-2y+z = 114x+2y+6z = 09x +7z = 11

Another Example

Now eliminate y using the second and third equations.Multiply each side of the second equation by 2.Add the result to the third equation.Now we have another linear equation in two variables.

2(2x+y+3z) = 2(0) 6x-2y-2z = 16 4x+2y+6z = 0 6x-2y-2z = 16 10x +4z = 16

9x+7z = 1110x+4z = 16

10(9x+7z) = 10(11)-9(10x+4z) = -9(16)

90x+70z = 110-90x-36z = -144 34z =-34

z =-1

Now we have two linear equations in two variables. Solve this system using any of the methods from the last section. (Elimination is used here)Eliminate x by multiplying the each side of the first equation by 10 and each side of the second equation by –9.

Add the two equations.The value of z is –1.

9x+7(-1)=11 9x-7=11 x=2

5(2)-2y+(-1)=11 10-2y-1=11

y=-1

By substituting the value of z into one of the equations in two variables, we can solve for the value of x.The value of x=2.

Finally, use one of the original equations to find the value of y.The value of y is –1.The answer can be written as an ordered triple (2,-1,-1)

Helpful WebsitesHelpful WebsitesSolving Systems with 3 variables: http://tutorial.math.lamar.edu/AllBrowsers/AlgebraTrigReview/SolveSystems.asp

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut50_systhree.htm

http://fp.academic.venturacollege.edu/rbrunner/3e_3s.htm

2-2 Self-Check Quiz: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-860861-9&chapter=2&lesson=2&quizType=1&headerFile=4&state=