Q2. - Algebra 2

Question

Write a system of equations that cannot be solved using Cramer’s Rule.

Step-by-Step Solution

Answer

The required system of equations is;

$\begin{array}{c}2m-n+3o=5\\ 3m+2n-5o=4\\ m-4y+11z=3\end{array}$

Step 1- Determine the approach that can be used

The formation of equations that returns the main determinant as 0 can be used.

Step 2 – Form the system of equations by using the approach and check them

Form the equations in such a manner so that the determinant that is in denominator of each variable is equal to 0.

$\begin{array}{c}2m-n+3o=5\\ 3m+2n-5o=4\\ m-4y+11z=3\end{array}$

Use Cramer’s rule for the variable m.

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

Use the provided equations and substitute the value of a, b, c, d, e, f, g, 

h, i, j, k and l into$m=\frac{\left|\begin{array}{ccc}j& b& c\\ k& e& f\\ l& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}$ after it perform simplification of

determinants to write the value of m.

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

The denominator of the value of m is 0 which means that the main determinant is 0 so each value will be infinite.

Step 3- Make the conclusion

As the solution represents that each variable has infinite value which means that the obtained system of equations cannot be solved with the help of Cramer’s rule.