please help me i have been stuck on this homework question for a very long time.

Question

Classify the system and identify the number of solutions.

4x + 3y + 6z = 25

x + 5y + 7z = 13

2x + 9y + z = 1

A.) consistent, dependent; infinite

B.) consistent, independent; one

C.) inconsistent; none

D.) consistent, independent; none

please just only give solution and explanation.

Answer 1

You need to see if you can solve for x, y and z without one of the equations being a scaled duplicate of another.

Turns out, by adding and subtracting and scaling rows to eliminate variables, x=4, y=-1 and z=2. So the system is independent and gives only one solution.

So the answer is B

Answer 2

step 1:

-4 * eq2 + eq1

-2*eq2 + eq3

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

-17y + -22z = -27

-y + -13z = -25

step 2: multplies everything by -1

17y + 22z = 27

y + 13z = 25

step 3: solves the 2x2 system by substitution

y=25 - 13z

17(25 - 13z) + 22z = 27

425 - 221z + 22z = 27

425 - 199z = 27

-199z = -398

z = 2

Then per step #2:

y + 13z = 25

y + 13(2) = 25

y + 26 = 25

y = -1

Finally per the last of the original 3 equations:

2x + 9y + z = 1

2x + 9(-1) + 2 = 1

2x + -9 + 2 = 1

2x + -7 = 1

2x = 8

x = 4

the solution is {x=4,y=-1,z=2}

It is a CONSISTENT, INDEPENDENT system

with ONE and ONLY ONE UNIQUE solution..

the answer is B