Chapter 8.5, Problem 72E

a.

To determine

State Cramer’s Rule for solving a system of linear equations.

Answer to Problem 72E

  $\boxed{\frac{\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}}$, $\boxed{\frac{\left|\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}}$ and $\boxed{\frac{\left|\begin{array}{ccc}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}}$

Explanation of Solution

Given information:

State Cramer’s Rule for solving a system of linear equations.

Calculation:

According to cramer’s rule,if

  $\begin{array}{l}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{2}x+b_{2}y+c_{2}z=d_2\\ a_{3}x+b_{3}y+c_{3}z=d_3\end{array}$

Represents a system of equations in three variables such that $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\ne 0$ . Then

  $x=\frac{D_X}{D}$

  $=\frac{\left|\begin{array}{ccc}{d}_1& b_1& c_1\\ d_2& b_2& c_2\\ d_3& b_3& c_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}$

  $x=\frac{D_y}{D}$

  $=\frac{\left|\begin{array}{ccc}{a}_1& d_1& c_1\\ a_2& d_2& c_2\\ a_3& d_3& c_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}$

  $x=\frac{D_z}{D}$

  $=\frac{\left|\begin{array}{ccc}{a}_1& b_1& d_1\\ a_2& b_2& d_2\\ a_3& b_3& d_3\end{array}\right|}{\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|}$

b.

To determine

Describe that the method you find easiest and most difficult to use.

Answer to Problem 72E

Cramer’s rule was the easiest method.

Guass-Jordan elimination was the difficult method.

Explanation of Solution

Given information:

At this point in the text, you have learned several methods for solving systems of linear equations. Briefly describe which method(s) you find easiest to use and which method(s) you find most difficult to use.

Calculation:

By using Gaussian elimination with back-substitution

By using Guass-Jordan elimination

By using inverse of the coefficient matrix

By using Cramer’s rule.

Out of these four methods we feel that Cramer’s rule was the easiest method, hence here we have to find the determinants only, we do not need to perform several roe operations to reduced matrix to echelon form. Also calculating determinants is quite easy than the finding inverse of the matrix.

Hence, we feel that Cramer’s rule was the easiest method.

Out of these four methods we feel that Guass-Jordan elimination was the difficult method, hence here we have to further reduced echelon form of the coefficient matrix to reduced echelon form.

Hence, we feel that Guass-Jordan elimination was the difficult method.