Question
Write a pair of linear equations which has the unique solution $x = -1, y = 3$. How many such pairs can you write$?$
✓
Answer
Condition for the pair of system to have unique solution
$\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}$
Let the equations are,
$a_1 x+b_1 y+c_1=0$
and $a_2 x+b_2 y+c_2=0$
Since, $x=-1$ and $y=3$ is the unique solution of these two equations, then
$a_1(-1)+b_1(3)+c_1=0 $
$\Rightarrow-a_1+3 b_1+c_1=0$
$ \text { and } a_2(-1)+b_2(3)+c 2=0 $
$ \Rightarrow-a_2+3 b_2+c_2=0$
So, the different valume of $a_1, a_2, b_1, b_2, c_1$ and $c_2$ satisfy the eqs. $(i)$ and $(ii).$
Hence, infinitely many pairs of linear equations are possible.
$\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}$
Let the equations are,
$a_1 x+b_1 y+c_1=0$
and $a_2 x+b_2 y+c_2=0$
Since, $x=-1$ and $y=3$ is the unique solution of these two equations, then
$a_1(-1)+b_1(3)+c_1=0 $
$\Rightarrow-a_1+3 b_1+c_1=0$
$ \text { and } a_2(-1)+b_2(3)+c 2=0 $
$ \Rightarrow-a_2+3 b_2+c_2=0$
So, the different valume of $a_1, a_2, b_1, b_2, c_1$ and $c_2$ satisfy the eqs. $(i)$ and $(ii).$
Hence, infinitely many pairs of linear equations are possible.
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