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] } = (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] } = ((-26)(2)-(-6)(13))/(-2)(2)-(-6)(5)) = (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] } = (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] }= ((-2)(13)-(5)(-26))/((-2)(2)-(-6)(5))= (104)/(26)=4`

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

Answer: Option A