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  • Can someone help me with this, Solve the system of equations x + 3y –2z = 0, 2x –y + 4z = 0, x –11y + 14z = 0.

Solve the system of equations

                        x + 3y –2z = 0, 2x –y + 4z = 0, x –11y + 14z = 0.

  • Option 1)

    x = -8k, y = 10k, and z = 7k

  • Option 2)

    x = 7k, y = 8k, and z = -10k

  • Option 3)

    x = 8k, y = -10k, and z = 7k

  • Option 4)

    x = -10k, y = 8k, and z = 7k

 

Answers (1)

best_answer

As we have learned

Solution of a non-homogeneous system of linear equations by matrix method -

If A is a singular matrix and adj(A).b=0 then the system of equations given by Ax=b has infinitely many solutions or no solution.

- wherein

 

 We have 

x + 3y –2z = 0,  2x –y + 4z = 0, x –11y + 14z = 0.

                        The given system of equations in the matrix form is written as

                        \begin{bmatrix} 1 &3 &-2 \\ 2 & -1 &4 \\ 1&-11 &14 \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}

                        i.e. AX = O                                                        …(1)

                        where A=\begin{bmatrix} 1 &3 &-2 \\ 2& -1& 4\\ 1& -11 & 14 \end{bmatrix},  X=\begin{bmatrix} x\\ y\\ z \end{bmatrix} and   O=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}

                        and | A | = 1 (-14 + 44) –3 (28 –4) (-2 (-22 + 1) = 30 –72 + 42 = 0

                        and therefore the system has non-trivial solutions. Now, we may write the first two of the given equations

                        x + 3y = 2z and 2x –y = -4z.

                        Solving these equations in terms of z, we get

                         x=-\frac{10}{7}z and   y=\frac{8}{7}z.

                        Putting x=-\frac{10}{7}z and y=\frac{8}{7}z  in the third equation of the given system,

                        we get   L.H.S=-\frac{10}{7}z \frac{88}{7}z\frac{88}{7}z+14z=0=RHS

                        Now if z = 7k, then x = -10k and y = 8k.

                        Hence, x = -10k, y = 8k, and z = 7k, where k is a real number.

 

 

 


Option 1)

x = -8k, y = 10k, and z = 7k

Option 2)

x = 7k, y = 8k, and z = -10k

Option 3)

x = 8k, y = -10k, and z = 7k

Option 4)

x = -10k, y = 8k, and z = 7k

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gaurav

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