Question

Determine the values of the parameter s for which the system has a unique solution, and describe the solution. x 1 - 5 sx 2

Answer 1

Answer:

$s \ne \±2$

$x_1 = \frac{3s - 2}{3(s^2 -4)}$

$x_2 = \frac{2(s- 6)}{5(s^2 - 4)}$

Step-by-step explanation:

Given

$3sx_1 +5x_2 = 3$

$12x_1 + 5sx_2 =2$

Required

Determine the value of s

Express the equations as a matrix

$A =\left[\begin{array}{cc}3s&5\\12&5s\end{array}\right]$

Calculate the determinant

$|A|= (3s*5s -5 *12)$

$|A|= (15s^2 -60)$

Factorize

$|A|= 15(s^2 -4)$

Apply difference of two squares

$|A|= 15(s -2)(s + 2)$

For the system to have a unique solution;

$|A| =0$

So, we have:

$15(s -2)(s+2) = 0$

Divide both sides by 15

$(s -2)(s+2) = 0$

Solve for s

$s -2 = 0\ or\ s +2 = 0$

$s = 2\ or\ s = -2$

The result can be combined as:

$s =\±2$

Hence, the system has a unique solution when $s \ne \±2$

Next, we solve for s using Cramer's rule.

We have:

$mat\ x_1 = \left[\begin{array}{cc}3&5\\2&5s\end{array}\right]$

Calculate the determinant

$|x_1| = (3 * 5s - 5 *2)$

$|x_1| = 15s - 10$

So:

$x_1 =\frac{|x_1|}{|A|}$

$x_1 = \frac{15s - 10}{15(s -2)(s+2)}$

Factorize

$x_1 = \frac{5(3s - 2)}{15(s -2)(s+2)}$

Divide by 5

$x_1 = \frac{3s - 2}{3(s -2)(s+2)}$

$x_1 = \frac{3s - 2}{3(s^2 -4)}$

Similarly:

$mat\ x_2 =\left[\begin{array}{cc}3s&3\\12&2\end{array}\right]$

Calculate the determinant

$|x_2| = 3s * 2 - 3 * 12$

$|x_2| = 6s- 36$

So:

$x_2 =\frac{|x_2|}{|A|}$

$x_2 = \frac{6s- 36}{15(s -2)(s+2)}$

Factorize

$x_2 = \frac{6(s- 6)}{15(s -2)(s+2)}$

Divide by 3

$x_2 = \frac{2(s- 6)}{5(s -2)(s+2)}$

$x_2 = \frac{2(s- 6)}{5(s^2 - 4)}$