System of Equations

Main Topic – System of Equations

Sub Topic - Determine whether a system of linear equations has no solutions (inconsistent system), one solution, or infinitely many solutions (dependent system).

Determine the type of solution for given system of equations.

2x + 3y + 4z = 9

3x + 5y -2z = 6

7x + 4y + z = 12

A. No solutions

B. Three solutions

C. Infinitely many solution

D. One solution

Answer: D

Rationale

First step is, to eliminate one variable from the first two equations.

Let us eliminate z first from the first two equations.

To eliminate z, multiply second equation by 2 and add it in the first equation.

   2x + 3y + 4z = 9

+ 6x + 10y - 4z = 12

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

   8x + 13y        = 21

Second step is to eliminate z from the last two equations.

Multiply third equation by 2 and then add it in the second equation.

     3x + 5y - 2z = 6

+ 14x + 8y + 2z = 24

    -------------------------

    17x + 13y      = 30

Next step is, to solve the system of equation with two variables which we obtained by eliminating z.

8x + 13y = 21

17x + 13y = 30

Elimination method is used:

   8x + 13y =21

 17x + 13y = 30 
 -       -          -
------------------------

   -9x           = -9

So, x = 1,

Use this value of x in any of the above equations.

8x + 13y = 21

=> 8*1 + 13y = 21 => 13y = 13

So y = 1

Now use the value of x and y in any of the original equation and solve for z.

2x + 3y + 4z = 9

=> 2*1 + 3*1 + 4z = 9

=> 5 + 4z = 9

=> 4z = 4

=> z = 1

Therefore the solution of system is x = 1, y = 1 and z = 1.

Since it has a unique solution, the type of solution is one solution.

Answer option A is incorrect. Eliminate z variable from the first two and the last two equations. Then solve the remaining system of equations with two variables by using the elimination method.

Answer option B is incorrect. Number of variables in the system of equations is three but the number of solutions is not the same as the number of variables.

Answer option C is incorrect. Eliminate z variable from the first two and the last two equations. Then solve the remaining system of equations with two variables by using the elimination method.