Question

Apply the Gaussian elimination process to determine the value of lembda for which the following linear system is consistent: x-3y+4=0, 3x-2y=lembda, y= 6-2x ?

Answer 1

Condition: Apply the Gaussian elimination process to determine the value of lambda for which the following linear system is consistent: $x - 3y + 4 = 0$ , $3x - 2y = \text{lambda}$ , $y = 6 - 2x$ ?

Solution:

Firstly, let us transform the form of the given equations to normal form.

$$\left\{ \begin{array}{l} x - 3 y = - 4 \\ 3 x - 2 y = \lambda \\ 2 x + y = 6 \end{array} \right.$$

Now, create augmented matrix of the given system.

$$\left( \begin{array}{c c c} 1 & - 3 & - 4 \\ 3 & - 2 & \lambda \\ 2 & 1 & 6 \end{array} \right)$$

Where the first column is coefficients of the variable $x$ , the second is coefficients of $y$ .

Then, using the Gaussian method of elimination, let us find our solution.

1) In the first step we compose the first linear equation and -3, then the result we add to the second linear equation. After we again compose the first linear equation and -2, then the result we add to the third linear equation.

The result of the first step is:

$$\left( \begin{array}{c c c} 1 & - 3 & - 4 \\ 0 & 7 & 12 + \lambda \\ 0 & 7 & 6 \end{array} \right)$$

2) In the second step we compose the third linear equation and -1, then we add result to the third second equation

The result of the second step is:

$$\left( \begin{array}{c c c} 1 & - 3 & - 4 \\ 0 & 0 & 6 + \lambda \\ 0 & 7 & 6 \end{array} \right)$$

3) Let us analyze the second equation. For doing that, rewrite it in the normal form of the equation:

$$0 * x + 0 * y = 6 + \lambda$$

So, this equation will be right if $\lambda = -6$ and because of that linear system will be consistent.

If take different value of $\lambda$, then we will get this $0 + 0 = k$, where $k \neq 0$. This means that there is no $x$ or $y$ that satisfy this equation. Because of that system will be inconsistent.

Answer: $\lambda = -6$;