Chapter 16.9, Problem 17WE

To determine

To prove: For the system of equations

  $\begin{array}{c}y=m_{1}x+b_1\\ y=m_{2}x+b_2\end{array}$

Thecondition $D\ne 0$ if and only if $m_1\ne m_2$ holds true.

Explanation of Solution

Given information:

The system of equations

  $\begin{array}{c}y=m_{1}x+b_1\\ y=m_{2}x+b_2\end{array}$

Proof:

Consider system of equations

  $\begin{array}{c}y=m_{1}x+b_1\\ y=m_{2}x+b_2\end{array}$

Write the above system in standard form,

  $\begin{array}{c}-m_{1}x+y=b_1\\ -m_{2}x+y=b_2\end{array}$

The determinant of above system of equations is,

  $\begin{array}{c}D=\left|\begin{array}{cc}-m_1& 1\\ -m_2& 1\end{array}\right|\\ =-m_1+m_2\end{array}$

If suppose $D\ne 0$ then,

  $\begin{array}{c}-m_1+m_2\ne 0\\ m_1\ne m_2\end{array}$

Next suppose $m_1\ne m_2$ so,

  $\begin{array}{c}{m}_1\ne m_2\\ -m_1+m_2\ne 0\\ D\ne 0\end{array}$

Hence, the condition $D\ne 0$ if and only if $m_1\ne m_2$ holds true for given system of equations.