<< Chapter < Page Chapter >> Page >

Solve the system of equations in three variables.

2 x + y −2 z = −1 3 x −3 y z = 5 x −2 y + 3 z = 6

( 1 , −1 , 1 )

Got questions? Get instant answers now!

Identifying inconsistent systems of equations containing three variables

Just as with systems of equations in two variables, we may come across an inconsistent system    of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as 3 = 7 or some other contradiction.

Solving an inconsistent system of three equations in three variables

Solve the following system.

        x −3 y + z = 4 ( 1 )   x + 2 y −5 z = 3 ( 2 ) 5 x −13 y + 13 z = 8 ( 3 )

Looking at the coefficients of x , we can see that we can eliminate x by adding equation (1) to equation (2).

      x −3 y + z = 4      ( 1 ) x + 2 y −5 z = 3      ( 2 )          y −4 z = 7      ( 4 )

Next, we multiply equation (1) by −5 and add it to equation (3).

5 x + 15 y 5 z = −20 ( 1 ) multiplied by −5 5 x 13 y + 13 z = 8 ( 3 ) ______________________________________               2 y + 8 z = −12 ( 5 )

Then, we multiply equation (4) by 2 and add it to equation (5).

−2 y 8 z = 14       ( 4 ) multiplied by 2 2 y + 8 z = 12    ( 5 ) _______________________________________ 0 = 2

The final equation 0 = 2 is a contradiction, so we conclude that the system of equations in inconsistent and, therefore, has no solution.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the system of three equations in three variables.

    x + y + z = 2          y −3 z = 1 2 x + y + 5 z = 0

No solution.

Got questions? Get instant answers now!

Expressing the solution of a system of dependent equations containing three variables

We know from working with systems of equations in two variables that a dependent system    of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.

Finding the solution to a dependent system of equations

Find the solution to the given system of three equations in three variables.

   2 x + y −3 z = 0 ( 1 ) 4 x + 2 y −6 z = 0 ( 2 )       x y + z = 0 ( 3 )

First, we can multiply equation (1) by −2 and add it to equation (2).

−4 x −2 y + 6 z = 0     equation  ( 1 ) multiplied by −2 4 x + 2 y −6 z = 0                    ( 2 ) ____________________________________________ 0 = 0

We do not need to proceed any further. The result we get is an identity, 0 = 0 , which tells us that this system has an infinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by −2 , and adding it to equation (1). We then perform the same steps as above and find the same result, 0 = 0.

When a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have

2 x + y −3 z = 0     x y + z = 0 _____________        3 x −2 z = 0

We then solve the resulting equation for z .

3 x −2 z = 0            z = 3 2 x

We back-substitute the expression for z into one of the equations and solve for y .

2 x + y 3 ( 3 2 x ) = 0       2 x + y 9 2 x = 0                         y = 9 2 x 2 x                         y = 5 2 x

So the general solution is ( x , 5 2 x , 3 2 x ) . In this solution, x can be any real number. The values of y and z are dependent on the value selected for x .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask