Which of the following are correct in respect of the system of equations $x + y + z = 8, x - y + 2 z = 6$ and $3 x - y + 5 z = k$ ?
1. They have no solution, if $k = 15$ .
2. They have infinitely many solutions, if $k = 20$ .
3. They have unique solution, if $k = 25$ .
Select the correct answer using the code given below

Question

Which of the following are correct in respect of the system of equations

x + y + z = 8, x - y + 2 z = 6

and

3 x - y + 5 z = k

?
1. They have no solution, if

k = 15

.
2. They have infinitely many solutions, if

k = 20

.
3. They have unique solution, if

k = 25

.
Select the correct answer using the code given below

Filo · Accepted Answer

First we need to know if the above equations are linearly dependent or no,

In order to figure that out lets find the determinant

D

which is;

D = ∣ ∣ 113 1 - 1 - 1 125 ∣ ∣ = 0

as

D = 0

the equations are linearly dependent, which means the system of equations can either be inconsistent or have infinitely many solutions. That shall depend on the value of k.

For there to be inifinitely many solutions , values of

D_{1}, D_{2}, D_{3}

should be

0

, defined as;

D_{1} = ∣ ∣ 86 k 1 - 1 - 1 125 ∣ ∣ D_{2} = ∣ ∣ 113 86 k 125 ∣ ∣ D_{3} = ∣ ∣ 113 1 - 1 - 1 86 k ∣ ∣

which on solving you get;

D_{1} = ∣ ∣ 86 k 1 - 1 - 1 125 ∣ ∣ = 3 k - 60 = 3 (k - 20) D_{2} = ∣ ∣ 113 86 k 125 ∣ ∣ = k - 20 D_{3} = ∣ ∣ 113 1 - 1 - 1 86 k ∣ ∣ = 40 - 2 k = - 2 (k - 20)

Now for all the Determinants to be zero , its evident that

k = 20

Hence statement 2 is correct.

Subsequently for there to be no solution

k \neq = 20

, hence statement 1 is correct as well,

As

D

is zero the system of equations CANNOT have a unique solution hence, statement 3 is WRONG.

Question Text	Which of the following are correct in respect of the system of equations $x + y + z = 8, x - y + 2 z = 6$ and $3 x - y + 5 z = k$ ? 1. They have no solution, if $k = 15$ . 2. They have infinitely many solutions, if $k = 20$ . 3. They have unique solution, if $k = 25$ . Select the correct answer using the code given below
Answer Type	Text solution:1
Upvotes	150

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Which of the following are correct in respect of the system of equations x+y+z=8,x−y+2z=6 and 3x−y+5z=k?1. They have no solution, if k=15.2. They have infinitely many solutions, if k=20.3. They have unique solution, if k=25.Select the correct answer using the code given below

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