Question

Use Cramer’s Rule to solve each system of equations.

$2x-y+3z=5$

$3x+2y-5z=4$

$x-4y+11z=3$

Short answer

The system of equation has no solutions.

Step-by-step solution

Step 1- Determine the value of x

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="1647355463064" $x=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$after it perform simplification of determinants to write the value of x.

$\begin{array}{c}x=\frac{\left|\begin{array}{ccc}5& -1& 3\\ 4& 2& -5\\ 3& -4& 11\end{array}\right|}{\left|\begin{array}{ccc}2& -1& 3\\ 3& 2& -5\\ 1& -4& 11\end{array}\right|}\\ =\frac{3}{0}\end{array}$

– Analyze the obtained value of x 

The denominator of the value of x is 0 which means that the main determinant is 0 so each value will become undefined.

Step 3- Make the conclusion

The obtained value of each value is undefined which means that there is no solution of the given system of equations.