Chapter 11.3, Problem 24AYU

$\{\begin{array}{l}-x+2y=5\\ 4x-8y=6\end{array}$

To determine

To find: The solution to the system of equations: $-x+2y=5$, $4x-8y=6$.

Answer to Problem 24AYU

Inconsistent system, no solutions.

Explanation of Solution

Given:

$-x+2y=5$, $4x-8y=6$

Formula used:

To find the value of a second order $2\times 2$ determinant:

$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=a\times d-b\times c=ad-bc$

Cramer’s rule states that:

The solution of a system of two linear equations $ax+by=m$ and $cx+dy=n$ is given by,

$x=\frac{D_x}{D};y=\frac{D_y}{D}$

Where we have,

$D=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|;D_x=\left|\begin{array}{cc}m& b\\ n& d\end{array}\right|$ and $D_y=\left|\begin{array}{cc}a& m\\ c& n\end{array}\right|$; provided $D=ad-bc\ne 0$.

When $D=0$, the equations are inconsistent or dependent.

Calculation:

$-x+2y=5$

$4x-8y=6$

Where we have:

$D=\left|\begin{array}{cc}-1& 2\\ 4& -8\end{array}\right|=8-8=0$

Since $D=0$, by the given concept the given system of equations is inconsistent and has no solutions.