Which classification best describes the following system of equations x-y-z=0 2x-y+2z=1 x-y+z=-2

(1) 3x+6y-12z=36
(2) x+2y-4z=12
(3) 4x+8y-16z=48

The first equation (1) is the second equation (2) multiplied by 3:
(2) x+2y-4z=12→3(x+2y-4z=12)→3x+6y-12z=36 (1)

The third equation (3) is the second equation (2) multiplied by 4:
(2) x+2y-4z=12→4(x+2y-4z=12)→4x+8y-16z=48 (3)

The equations are linearly dependent, Then the system of equations is dependent, and then consistent too.

Inconsistent and Independent

Explanation:

EQ 1: 5x + 10y + 15z = 60

EQ 2: 2x + 4y + 6z = 60

EQ 3:  x + 2y + 3z = 60

EQ 1: 5x + 10y + 15z = 60  → -2(5x + 10y + 15z = 60) → -10x - 20y - 30z = -120

EQ 2: 2x + 4y + 6z = 60   →   5(2x +  4y +  6z  = 60)  → 10x + 20y + 30z = 300

                                                                                                             0   = 180

                                                                                                             FALSE

False statement means EQ1 and EQ2 are parallel lines.

EQ 2: 2x + 4y + 6z = 60 → -1(2x + 4y + 6z = 60) → -2x - 4y - 6z = -60

EQ 3:  x + 2y + 3z = 60  → 2( x + 2y + 3z = 60)  →  2x + 4y + 6z = 120

                                                                                                0   = 60

                                                                                                 FALSE

False statement means EQ2 and EQ3 are parallel lines.

EQ1 || EQ2 and EQ2 || EQ3, so EQ1 || EQ3

Since all lines are parallel to each other there will be no solution.

Inconsistent because the lines do no not cross

Independent because they do not rely on another equation  

The correct option is option third, i.e., consistent and independent.

Explanation:

The system of equation have three equation,

            ..... (1)

         ..... (2)

    .... (3)

Put the value of x and y from equation (1) and (2) in equation (3).

The given system of equation have a unique solution (5,6,11).

The system of equation is inconsistent if  there is no solution and consistent if there exist at least one solution. If the solution is unique the it is independent and if there are more than one solutions then it is dependent.

Since in above calculation we find the values of (x,y,z) and we get a unique value for each variable, it means the solution is unique and the system of linear equation is consistent and independent.

Therefore the option third is correct.

These three planes are consistent and independent.

Explanation:

Since,  if the system of planes has a solution then it is  called Consistent, While, if it does not have any solution then it is called inconsistent.

Further, If the consistent system has infinite solution then it is dependent but if it has only a unique solution then it is called independent.

Here, given equations of planes are

12x+5y-3z=36           (1 )

x-2y+4z=3                 (2)

9x-10y+5z=27          (3)

From equation (1), 3z=12x+5y-36⇒z=4x+5y/3-12   (4)

after putting this value in equation (2) and (3), we will get two equation in variables x and y

So, x-2y+4(4x+5y/3-12)=3 ⇒x-2y+16x+20y/3-48=3⇒3x-6y+48x+20y-144=9⇒51x+14y=153   (5)

And, 9x-10y+5(4x+5y/3-12)=27⇒ 9x-10y+20x+25y/3-60=27⇒ 27x-30y+60x+25y-180=81⇒87x-5y=261   (6)

after solving equation equation (5) and (6) we will get x=3 and y=0

substituting these values in equation (4) we will get z=0

Thus the solution of these three plane (1), (2) and (3) is x=3, y=0 and z=o

Which is the unique solution, Thus the given planes are consistent and independent.

Consistent and dependent

Step-by-step explanation:

We are given the following equation:

1.

2.

3.

For equation 1 and 3, if we take out the common factor (3 and 4 respectively) out of it then we are left with which is the same as the equation number 2.

There is at least one set of the values for the unknowns that satisfies every equation in the system and since there is one solution for each of these equations, this system of equations is consistent and dependent.

c would have to be right.

consistent and independent

Step-by-step explanation:

its d

The correct option is c.

Step-by-step explanation:

The given equations are

                 .... (1)

            ..... (2)

              .... (3)

Subtract equation (3) form equation (1).

The value of z is -1. Put this value in equation (1) and (2).

               .... (4)

           ..... (5)

Subtract equation (5) from equation (4).

The solution of given system of equations is (4,5,-1). It is a unique solution.

Therefore the given system of equations is consistent and independent. Option c is correct.

The given system of equations is consistent and independent so, is correct.

Further explanation:

A system of equations is said to be a consistent system if the solution exists and if the solution does not exist then it is an inconsistent system.  

If a consistent system has a unique solution then it is an independent system of equations but if the number of solutions is infinite then it is a dependent system of equations.

Label the given equations as shown below:

              …… (1)

        …… (2)

        …… (3)

The augmented matrix for the above equations is, as follows:

Apply row transformation and as,

 

Now, apply row transformation and as,

 

Now, apply row transformations and as,

 

The equations obtained from the above augmented matrix are,

 

The first equation is simplified to obtain the value of z as,

 

Substitute for in the equation  to obtain the value of as,

Now, substitute for and for in the equation to obtain the value of as,

 

Therefore, the value of is , the value of is and the value of is and the system has a unique solution.

Thus, the given system of equations is consistent and independent.

Option (1)

Here, the first option is inconsistent and dependent.

There exists a solution of the given system of equations so it is consistent.

Therefore, option (1) is incorrect.

Option (2)

Here, the second option is consistent and dependent.

There exists a solution of the given system of equations so it is consistent.

Also, the solution is unique therefore the system of equations is independent.

Therefore, option (2) is incorrect.

Option (3)

Here, the third option is consistent and independent.

There exists a solution of the given system of equations so it is consistent.

Also, the solution is unique therefore the system of equations is independent.

Therefore, option (3) is correct.

Option (4)

Here, the fourth option is inconsistent and independent.

There exists a solution of the given system of equations so it is consistent.

Therefore, option (4) is incorrect.

Learn more:

1. A problem on circle

2. A problem on general equation of a circle

Answer details

Grade: High school

Subject: Mathematics

Chapter: System of linear equations

Keywords: Equations, unique solution, independent, dependent, consistent, inconsistent, infinite solutions, homogeneous equation, non- homogeneous equation, determinant.