91Ó°ÊÓ

Apply the Cramer’s rule for the value of x.

$x=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the provided equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into role="math" localid="1647257774250" $x=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{ccc}−32& 2& −3\\ 54& −3& 1\\ 78& 2& 8\end{array}\right|}{\left|\begin{array}{ccc}4& 2& −3\\ −1& −3& 1\\ 0& 2& 8\end{array}\right|}\\ =\frac{−902}{−82}\\ =11\end{array}$

Apply the Cramer’s rule for the value of y.

$y=\frac{\left|\begin{array}{ccc}a& j& c\\ d& k& f\\ g& l& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the provided equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into localid="1647257906552" $y=\frac{\left|\begin{array}{ccc}a& j& c\\ d& k& f\\ g& l& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of y.

$\begin{array}{c}y=\frac{\left|\begin{array}{ccc}4& −32& −3\\ −1& 54& 1\\ 0& 78& 8\end{array}\right|}{\left|\begin{array}{ccc}4& 2& −3\\ −1& −3& 1\\ 0& 2& 8\end{array}\right|}\\ =\frac{1394}{−82}\\ =−17\end{array}$

$\begin{array}{c}y=\frac{\left|\begin{array}{ccc}4& −32& −3\\ −1& 54& 1\\ 0& 78& 8\end{array}\right|}{\left|\begin{array}{ccc}4& 2& −3\\ −1& −3& 1\\ 0& 2& 8\end{array}\right|}\\ =\frac{1394}{−82}\\ =−17\end{array}$

Apply the Cramer’s rule for the value of z.

$z=\frac{\left|\begin{array}{ccc}a& b& j\\ d& e& k\\ g& h& l\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$

Use the provided equations and substitute the value of a, b, c, d, e, f, g, h, i, j, k and l into role="math" localid="1647258065943" $m=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$after it perform simplification of determinants to write the value of z.

$\begin{array}{c}y=\frac{\left|\begin{array}{ccc}4& 2& −32\\ −1& −3& 54\\ 0& 2& 78\end{array}\right|}{\left|\begin{array}{ccc}4& 2& −3\\ −1& −3& 1\\ 0& 2& 8\end{array}\right|}\\ =\frac{−1148}{−82}\\ =14\end{array}$

The obtained value of x is 11, y is $-17$ and z is 14 so the solution is $\left(11,-17,14\right)$.