Chapter 7, Problem 134CR

To determine

To find: The solution of the given system of equations.

Answer to Problem 134CR

The given system of equations is inconsistent and hence Cramer’s rule cannot be applied to find the solution.

Explanation of Solution

Given: The given system of equations are-

  $\begin{array}{l}8x-4y=3\\ -12x+6y=-15\end{array}$

Concept Used: Consider the system of equations-

  $a_{1}x+b_{1}y=c_1$

  $a_{2}x+b_{2}y=c_2$

The associated coefficient matrix is-

  $A=\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]$

Consider the following determinants-

  $D=\left|\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right|$ (Determinant of the coefficient matrix)

  $D_x=\left|\begin{array}{cc}{c}_1& b_1\\ c_2& b_2\end{array}\right|$ (Determinant of the coefficient matrix where the coefficients of x are replaced by the constant terms)

  $D_y=\left|\begin{array}{cc}{a}_1& c_1\\ a_2& c_2\end{array}\right|$ (Determinant of the coefficient matrix where the coefficients of y are replaced by the constant terms)

Hence, according to Cramer’s rule the solution of the given system of equations is given by-

  $x=\frac{D_x}{D}$ and $y=\frac{D_y}{D}$ , provided $D\ne 0$ .

If however $D=0$ then the given system is inconsistent and Cramer’s rule can’t be applied to find a solution.

Calculations: The given system of equations are-

  $\begin{array}{l}8x-4y=3\\ -12x+6y=-15\end{array}$

The associated coefficient matrix is $\left[\begin{array}{cc}8& -4\\ -12& 6\end{array}\right]$ .

Thus,

  $D=\left|\begin{array}{cc}8& -4\\ -12& 6\end{array}\right|=48-48=0$

Since, $D=0$ therefore the given system of equations is inconsistent and hence the solution cannot be acquired using Cramer’s rule.