Q. 44

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$2 x + y - 3 z = 0 - 2 x + 2 y + z = - 7 3 x - 4 y - 3 z = 7$

Verified

Answer

The solution of the system of equation is $x = \frac{56}{13}, y = - \frac{7}{13}, z = \frac{35}{13}$

1Step 1: Given information

We are given a system of equations

$2 x + y - 3 z = 0 - 2 x + 2 y + z = - 7 3 x - 4 y - 3 z = 7$

2Step 2: Multiply equation 2 by 3

We get,

$- 6 x + 6 y + 3 z = - 21$ (4)

Now add equation 1 and 4

We get,

$2 x + y - 3 z = 0 - 6 x + 6 y + 3 z = - 21 - 4 x + 7 y = - 21$

Hence $- 4 x + 7 y = - 21$

Now we add equation 4 and 3

We get,

$3 x - 4 y - 3 z = 7 - 6 x + 6 y + 3 z = - 21 - 3 x + 2 y = - 14$

We get, $- 3 x + 2 y = - 14$

3Step 3: Now we solve equation 4 and 5

We get,

$- 12 x + 21 y = - 63 + 12 x - 8 y = 56 13 y = - 7$

Hence we get $y = - \frac{7}{13}$

4Step 4: Find the value x and z

We get,

$3 x = 2 y + 14 3 x = 2 (- \frac{7}{13}) + 14 3 x = - \frac{14}{13} + 14 3 x = \frac{168}{13} x = \frac{56}{13}$

Similarly

$2 x + y - 3 z = 0 3 z = 2 x + y 3 z = 2 (\frac{56}{13}) - \frac{7}{13} 3 z = \frac{105}{13} z = \frac{35}{13}$

5Step 5: Conclusion

The solution of the equation are $x = \frac{56}{13}, y = \frac{- 7}{13}, z = \frac{35}{13}$