Question

Given the system of equations 2 x − 3 y − 9 z = 9 x + 3 z = 3 − 3 x + y − 4 z = − 10 (a) determine whether the system is inconsistent or dependent; Your answer is (input inconsistent or dependent) dependent Correct (b) if your answer is dependent in (a), find the complete solution. Write x and y as functions of z .

Answer 1

Dependent

`x=3-3z`

`y=-5z-1`

Explanation:

We are given that system of equations

`2x-3y-9z=9` (Equation I)

`x+3z=3` ( Equation II)

`-3x+y-4z=-10` (Equation III)

Equation III multiply by 3 and then add to equation I then we get

`-7x-21z=-21`

Divided by -7 then we get

`x+3z=3`

Now , obtained equation subtract from equation II then we get

0=0

Hence, equation II is a linear combination of equation I and equation III.

Therefore, system is dependent because system has infinite solutions.

`x=3-3z`

Substitute the value of x in equation I then we get

`2(3-3z)-3y-9z=9`

`6-6z-3y-9z=9`

`3y=-15z+6-9=-15z-3`

`y=-5z-1`