Question

Use Cramer's Rule to solve the following system: –2x – 6y = –26 5x + 2y = 13

Answer 1

We have the linear system:

$\left \{ {{-2x-6y=-26} \atop {5x+2y=13}} \right.$

which in Matrix format is

$\left[\begin{array}{ccc}a_1&b_1\\a_2&b_2\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}c_1\\c_2\end{array}\right]$

$\left[\begin{array}{ccc}-2&-6\\5&2\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}-26\\13\end{array}\right]$

We then find the value of x and y use the Cranmer's Rule:

$x= \frac{ \left[\begin{array}{ccc}c_1&b_1\\c_2&b_2\end{array}\right] }{ \left[\begin{array}{ccc}a_1&a_2\\b_1&b_2\end{array}\right] } = \frac{c_1b_2-b_1c_2}{a_1b_2-a_2b_1}$

$x= \frac{ \left[\begin{array}{ccc}-26&-6\\13&2\end{array}\right] }{ \left[\begin{array}{ccc}-2&-6\\5&2\end{array}\right] } = \frac{(-26)(2)-(-6)(13)}{-2)(2)-(-6)(5)} = \frac{26}{26}=1$

$y= \frac{ \left[\begin{array}{ccc}a_1&c_1\\a_2&c_2\end{array}\right] }{ \left[\begin{array}{ccc}a_1&b_1\\a_2&b_2\end{array}\right] } = \frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$

$y= \frac{ \left[\begin{array}{ccc}-2&-26\\5&13\end{array}\right] }{ \left[\begin{array}{ccc}-2&-6\\5&2\end{array}\right] }= \frac{(-2)(13)-(5)(-26)}{(-2)(2)-(-6)(5)}= \frac{104}{26}=4$

So we have the answers:
x = 1 and y = 4

Answer: Option A