Chapter 8.5, Problem 9E

To determine

To Calculate: The solution of the given system of equations using Cramer’s rule

Answer to Problem 9E

The given system of equations has no solution or infinitely many solutions.

Explanation of Solution

Given information:

The given system of equations is

  $\begin{array}{l}3x+2y=-2\\ 6x+4y=4\end{array}$

Concept and Formula Used:

We use Cramer’s rule.

Cramer’s rule: If a system of n linear equations in n variables has a coefficient matrix A with a non-zero determinant $\left|A\right|$ , then the solution of the system is

  $x_1=\frac{\left|A_1\right|}{\left|A\right|},x_2=\frac{\left|A_2\right|}{\left|A\right|},\mathrm{.......},x_n=\frac{\left|A_n\right|}{\left|A\right|}$

where ith column of $A_i$ is the column of constants of the system of equations. If the determinant is zero, then the system of equations has either no solution or infinitely many solution.

Calculation:

The given system of equations is

  $\begin{array}{l}3x+2y=-2\\ 6x+4y=4\end{array}$

So, the coefficient matrix is $A=\left[\begin{array}{cc}3& 2\\ 6& 4\end{array}\right]$ and the column matrix of constants is $b=\left[\begin{array}{l}-2\\ 4\end{array}\right]$ . The determinant of coefficient matrix is

  $\left|A\right|=\left|\begin{array}{cc}3& 2\\ 6& 4\end{array}\right|=\left(3\right)\left(4\right)-\left(2\right)\left(6\right)=12-12=0$ .

As the determinant is zero, so the given system of equations has either no solution or infinitely many solutions.

Conclusion:

The given system of equations has no solution or infinitely many solutions.