# Class 12 RD Sharma Solutions – Chapter 8 Solution of Simultaneous Linear Equations – Exercise 8.2

### Question 1.

2x â€“ y + z = 0

3x + 2y â€“ z = 0

x + 4y + 3z = 0

Solution:

Given

2x â€“ y + z = 0

3x + 2y â€“ z = 0

X + 4y + 3z = 0

The system can be written as

A X = 0

Now, |A| = 2(6 + 4) + 1(9 + 1) + 1(12 â€“ 2)

|A| = 2(10) + 10 + 10

|A| = 40 â‰  0

Since, |A|â‰  0, hence x = y = z = 0 is the only solution of this homogeneous equation.

### Question 2.

2x â€“ y + 2z = 0

5x + 3y â€“ z = 0

X + 5y â€“ 5z = 0

Solution:

Given 2x â€“ y + 2z = 0

5x + 3y â€“ z = 0

X + 5y â€“ 5z = 0

A X = 0

Now, |A| = 2(â€“ 15 + 5) + 1(â€“ 25 + 1) + 2(25 â€“ 3)

|A| = â€“ 20 â€“ 24 + 44

|A| = 0

Thus, the system has infinite solutions

Let z = k

2x â€“ y = â€“ 2k

5x + 3y = k

### Question 3.

3x â€“ y + 2z = 0

4x + 3y + 3z = 0

5x + 7y + 4z = 0

Solution:

Given:

3x â€“ y + 2z = 0

4x + 3y + 3z = 0

5x + 7y + 4z = 0

A X = 0

Now, |A| = 3(12 â€“ 21) + 1(16 â€“ 15) + 2(28 â€“ 15)

|A| = â€“ 27 + 1 + 26

|A| = 0

Hence, the system has infinite solutions

Let z = k

3x â€“ y = â€“ 2k

4x + 3y = â€“ 3k

### Question 4.

x + y â€“ 6z = 0

x â€“ y + 2z = 0

â€“ 3x + y + 2z = 0

Solution:

Given:

x + y â€“ 6z = 0

x â€“ y + 2z = 0

â€“ 3x + y + 2z = 0

A X = 0

Now, |A| = 1(â€“ 2 â€“ 2) â€“ 1(2 + 6) â€“ 6(1 â€“ 3)

|A| = â€“ 4 â€“ 8 + 12

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = 6k

x â€“ y = â€“ 2k

### Question 5. Solve the system of homogeneous linear equations by matrix method:

x + y + z = 0

x â€“ y â€“ 5z = 0

x + 2y + 4z = 0

Solution:

Given:

x + y + z = 0

x â€“ y â€“ 5z = 0

x + 2y + 4z = 0

A X = 0

Now, |A| = 1(6) â€“ 1(9) + 1(3)

|A| = 9 â€“ 9

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = â€“k

x â€“ y = 5k

### Question 6. Solve the system of homogeneous linear equations by matrix method:

x + y â€“ z = 0

x â€“ 2y + z = 0

3x + 6y â€“5z = 0

Solution:

Given:

x + y â€“ z = 0

x â€“ 2y + z = 0

3x + 6y â€“5z = 0

A X = 0

Now, |A| = 1(4) â€“ 1(â€“8) â€“ 1(12)

|A| = 4 + 8  â€“ 12

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = â€“k

x â€“ 2y = â€“k

### Question 7. Solve the system of homogeneous linear equations by matrix method:

3x + y â€“ 2z = 0

x + y + z = 0

x â€“ 2y + z =0

Solution:

Given:

3x + y â€“ 2z = 0

x + y + z = 0

x â€“ 2y + z =0

A X = 0

Now, |A| = 3(3) â€“ 1(0) â€“ 2(â€“3)

|A| = 9 â€“ 0  + 6

|A| = 15 â‰  0,

Hence, the given system has only trivial solutions given by x = y = z = 0.

### Question 8. Solve the system of homogeneous linear equations by matrix method:

2x + 3y â€“ z =0

x â€“ y â€“ 2z = 0

3x + y + 3z = 0

Solution:

Given:

2x + 3y â€“ z =0

x â€“ y â€“ 2z = 0

3x + y + 3z = 0

A X = 0

Now, |A| = 2(â€“3 + 2) â€“ 3(3 + 6) â€“ 1(4)

|A| = â€“2 â€“ 27  â€“ 4

|A| = â€“33 â‰  0,

Hence, the given system has only trivial solutions given by x = y = z = 0.

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