Question

Which of the following systems of equations has a unique solution?
1) $3x -y =2 , 6x -2y =3$
2) $2x - 5y = 3, 6x - 15y =9$
3) $x -2y = 3, 3x - 2y =1$
4) $2x -3y =4 , 4x - 6y =8$

Answer 1

Hint: By unique solution of equations we mean that the equations intersect only at one point when drawn on Cartesian coordinates or solved by keeping the value of one variable into another equation.
Using the above concept we will check out the equations which have a unique solution.

Complete step-by-step solution:
We will check one by one each option and then will conclude whether the solution of the equation is unique or not.
We have equations;
$3x -y =2$ ................(1)
$6x -2y =3$...............(2)
On subtracting equation 1 and 2 by equating coefficients of any one variable we have,
$\Rightarrow (3x - y = 2) \times 2 $
$ \Rightarrow (6x - 2y = 3) \times 1 $
(we have equalized the y variable)
$ \Rightarrow 0 = 1$ (will come out when the two equations are subtracted)
It means the solution of the equations is infinite.
Similarly, we have
$2x - 5y = 3$ ..............(3)
$6x - 15y =9$..............(4)
On subtracting equation 3 and 4 by equalizing the coefficient of any one variable;
$\Rightarrow (2x - 5y = 3) \times 6 $
$ \Rightarrow (6x - 15y = 9) \times 2 $
(We have equalized x variable )
$ \Rightarrow y = 0$ comes out as the solution which means the solution is infinite.
Next set of equations we have;
$x -2y = 3$..............(5)
$3x - 2y =1$............(6)
On subtracting the equation 5 and 6 by equalizing the coefficient of any one variable;
Coefficient of y is equal in equations 5 and 6, so we can directly subtract equation 5 and 6.
We have x = 1 which the equations have a unique solution.
Last set of equations we have;
$2x -3y =4$ ................(7)
$4x - 6y =8$...............(8)
On subtracting equation 7 and 8 by equalizing the coefficient of any one variable;
$\Rightarrow \left( {2x - 3y = 4} \right) \times 4 $
$ \Rightarrow \left( {4x - 6y = 8} \right) \times 2 $ (we have equalized x variable)
After subtraction we have
0=0 which means the solution of the equation has infinite solution.

Thus, option 3 is the correct answer.

Note: We have another method of calculating the unique solution of the equation, which is finding the ratio of coefficient of x and y , if the ratio of coefficient of x and y is equal then the solution is not unique, if the ratio of coefficient of x and y is not equal then the solution of the equation is unique.