Question

Show that each of the following systems of equations has a unique solution and solve it:

$3x+5y=12,5x+3y=4.$

Answer 1

$3x+5y–12=0,$

$5x+3y–4=0$

$a_1=3,b_1=5,c_1=-12$

$a_2=5,b_2=3,c_2=-4$

Thus,

$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

Hence, the given system of equations has a unique solution.

The given equations are,

$3x+5y=12$.....(1)

$5x+3y=4$.......(2)

Multiplying (1) by 3, and (2) by 5, we get

$9x+15y=36$......(3)

$25x+15y=20$......(4)

Subtracting (3) from (4), we get

$16x=-16$

$x=\frac{-16}{16}=-1$

$x=-1$

Putting x = -1 in (3), we get

$\left(9\right)(-1)+15y=36$

$-9+15y=36$

$15y=36+9$

⇒ $y=\frac{45}{15}=3$

Hence, the solution is $x=-1,y=3$