Question

Write a pair of linear equations which has the unique solution $x=-1,y=3$. How many such pairs can you write?

Answer 1

Find pair of linear equations which has a unique solution $x=-1,y=3$ and find number of such pairs :

Take the equations as follows:

$\begin{array}{rcl}{a}_{1}x+b_{1}y+c_1& =& 0\\ a_{2}x+b_{2}y+c_2& =& 0\end{array}$

As $x=-1,y=3$ is a solution of each of the equations,

$\begin{array}{rcl}-a_1+3b_1+c_1& =& 0\\ -a_2+3b_2+c_2& =& 0\end{array}$

For different values of $a_1,b_1,c_1,a_2,b_2,c_2$, we get different equations.

So, there exist many such pairs of equations which has a unique solution $x=-1,y=3$.

Take $a_1=1,b_1=1,c_1=2,a_2=2,b_2=3,c_2=1$

$\begin{array}{rcl}x+y+2& =& 0\\ 2x+3y+1& =& 0\end{array}$

Hence, there exist infinitely many pairs of equations which has a unique solution $x=-1,y=3$. Also, unique pair of equations is $\begin{array}{rcl}x+y+2& =& 0,2x+3y+1=0\end{array}$