Question

Find the number of solutions for the system of equations
4x + y + 2z = 5
x - 5y + 3z = 10
9x - 3y + 7z = 20

Answer 1

4x + y + 2z = 5 … (1)
x - 5y + 3z = 10 $\to$ x = 10 - 5y - 3z
9x - 3y + 7z = 20 … (2)
Eqn (1) becomes, 4(10 - 5y - 3z) + y + 2z = 5
40 - 20y - 12z + y + 2z = 5
-19y -10z = -35
19y + 10z = 35 … (3)
Eqn (2) becomes 9(10 - 5y - 3z) - 3y + 7z = 20
$\therefore$ 90 - 45y - 27z - 3y + 7z = 20
- 42y - 20z = - 70
21y + 10z = 35 … (4)
From (3) & (4),
$\frac{19}{21}\ne \frac{10}{10}=\frac{35}{35}$ s/m of eqn has unique solution.
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}=\frac{c_1}{c_2}\to$ Unique solution. $\frac{a_1}{a_2}\frac{b_1}{b_2}\ne \frac{c_1}{c_2}\to$ no solution.
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\to$ Infinite solution.