Question

Show that the following system of equations has a unique solution:
2x - 3y = 17,
4x + y = 13.
Also, find the solution of the given system of equations.

Answer 1

The given system of equations is:
2x - 3y - 17 = 0 ….(i)
4x + y - 13 = 0 …..(ii)
The given equations are of the form
`a_1x+b_1y+c_1 = 0 and a_2x+b_2y+c_2 = 0`
where, `a_1 = 2, b_1= -3, c_1= -17 and a_2 = 4, b_2 = 1, c_2= -13`
Now,
`(a_1)/(a_2) = 2/4 = 1/2 and (b_1)/(b_2) = (−3)/1 = -3`
Since, `(a_1)/(a_2) ≠ (b_1)/(b_2)`, therefore the system of equations has unique solution.
Using cross multiplication method, we have

`x/(b_1c_2− b_2c_1) = y/(c_1a_2− c_2a_1) = 1/(a_1b_2− a_2b_1)`
`⇒ x/(−3(−13)−1×(−17)) = y/(−17 ×4−(−13)×2) = 1/(2 ×1−4×(−3))`
`⇒ x/(39+17) = y/(−68+26) = 1/(2+12)`
`⇒ x/56 = y/(−42) = 1/14`
`⇒ x = 56/14, y = (−42)/14`
⇒ x = 4, y = -3
Hence, x = 4 and y = -3.