+0  
 
0
315
2
avatar+1278 

For a certain value of k, the system

x + y + 3z = 10,

-4x + 2y +5z = 7,

k*x + z = 3

has no solutions. What is this value of k?

AWESOMEEE  Aug 11, 2015

Best Answer 

 #1
avatar+17655 
+10

The third equation has no y-term, so let's eliminate the y-term from the combination of the first two equations.

x + y + 3z  =  10     --->     x -2     --->     -2x - 2y - 6z  =  -20

-4x + 2y + 5z  =  7       --->                      -4x + 2y + 5z  =  7

Add down the columns:                             -6x - z  =  -13   

Since  kx + z  =  3        --->         z  =  3 - kx

Substituting these:        -6x - (3 - kx)  =  -13

                                    -6x - 3 + kx  =  -13

                                      6x + 3 - kx  =  13

                                          6x - kx  =  10

                                          x(6 - k)  =  10

If  6 - k  =  0,  there is no value for x that will result in a product of 10,

so  k = 6  results in no possible solution.

For every other value of k, there will be a value of x that can produce a product of 10.

geno3141  Aug 12, 2015
Sort: 

2+0 Answers

 #1
avatar+17655 
+10
Best Answer

The third equation has no y-term, so let's eliminate the y-term from the combination of the first two equations.

x + y + 3z  =  10     --->     x -2     --->     -2x - 2y - 6z  =  -20

-4x + 2y + 5z  =  7       --->                      -4x + 2y + 5z  =  7

Add down the columns:                             -6x - z  =  -13   

Since  kx + z  =  3        --->         z  =  3 - kx

Substituting these:        -6x - (3 - kx)  =  -13

                                    -6x - 3 + kx  =  -13

                                      6x + 3 - kx  =  13

                                          6x - kx  =  10

                                          x(6 - k)  =  10

If  6 - k  =  0,  there is no value for x that will result in a product of 10,

so  k = 6  results in no possible solution.

For every other value of k, there will be a value of x that can produce a product of 10.

geno3141  Aug 12, 2015
 #2
avatar+78618 
+5

 

 

Here's another way to tackle this one - although I still probably like geno's method better !!!!

 

Get y by itself in the first equation

 

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

 

Substitute this into the second equation and simplify:

 

-4x + 2y +5z = 7    →   -4x + 2(10 - x - 3z) + 5z = 7   →   -4x + 20 -2x - 6z + 5z  = 7  →  -6x - z = -13  →

 

6x + z  = 13

 

So we have this system:

 

k*x + z = 3

 

6x + z = 13 

 

And notice that, if k =6, we would have this system:

 

6x + z  = 3      and

 

6x + z  = 13       but this is impossible, because two things added together can't possibly have two different outcomes 

 

So....as geno found, when k = 6, the system has no solution...!!!

 

 

  

CPhill  Aug 12, 2015

10 Online Users

avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details