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# System

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For a certain value of k, the system

x + y + 3z = 10

-4x + 8y + 5z = 7

kx + 2z = 3

has no solutions. What is this value of k?

Jan 10, 2022

#1
+13036
+1

What is this value of k?

Hello Guest!

x +  y   + 3z  = 10   | *5

-4x + 8y + 5z  = 7     | *3

kx           + 2z = 3

5x +5y +15z = 50

-12x+24y+15z = 21

-------------------------

17x -19y         = 29      | *2

x +  y   + 3z      = 10   | *2

kx         + 2z      =  3    | *3

2x + 2y  +6z      = 20

3kx         +6z      = 9

--------------------------

(2 - 3k)x +   2y = 11     | *19

(38 - 57k)x + 38y =  209

34x             - 38y =   58

-------------------------------

(38 - 23k)x           = 267   | :23

(1.652 - k)x          =11.609

$$x=\dfrac{11.609}{1.652-k}$$

$$y=\frac{34x-58}{38}=\frac{\frac{34\cdot 11.609}{1.652-k}-58}{38}$$

$$y=\dfrac{10.387}{1.652-k}-1.526$$

$$z=\frac{3-kx}{2}=\frac{ 3-\frac{11.609k}{1.652-k} }{2}$$

$$z=1.5-\dfrac{5.8045k}{1.652-k}$$

$$\large k\in \mathbb R$$ | -$$\infty$$ < k < $$\infty$$

!

Jan 10, 2022
edited by asinus  Jan 10, 2022
edited by asinus  Jan 10, 2022