Question 576307
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For the matrix


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left[a_1\ \ b_1\cr a_2\ \ b_2\right]]


The determinant is


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ D\ =\ \left|a_1\ \ b_1\cr a_2\ \ b_2\right|\ =\ a_1b_2\ -\ a_2b_1]


Given the system


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left\{a_1x\ +\ b_1y\ =\ c_1\cr a_2x\ +\ b_2y\ =\ c_2\right]


First calculate the determinant of the coefficient matrix:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ D\ =\ \left|a_1\ \ b_1\cr a_2\ \ b_2\right|\ =\ a_1b_2\ -\ a_2b_1]


Then replace the first column with the constant values and calculate *[tex \LARGE D_x]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ D_x\ =\ \left|c_1\ \ b_1\cr c_2\ \ b_2\right|\ =\ c_1b_2\ -\ c_2b_1]


Then replace the second column with the constant values and calculate *[tex \LARGE D_y]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ D_y\ =\ \left|a_1\ \ c_1\cr a_2\ \ c_2\right|\ =\ a_1c_2\ -\ a_2c_1]


Finally, solve for *[tex \LARGE x\ =\ \frac{D_x}{D}] and *[tex \LARGE y\ =\ \frac{D_y}{D}]


For your problem:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ D\ =\ \left(\frac{2}{3}\right)\left(\frac{5}{2}\right)\ -\ \left(-\frac{1}{4}\right)\left(-\frac{5}{2}\right)\ =\ \frac{5}{3}\ -\ \frac{5}{8}\ =\ \frac{25}{24}]


I'll let you do the rest of the arithmetic.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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