For real numbers \(\alpha\) and \(\beta\), consider the following system of linear equations: \(x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta\). If the system has infinite solutions, then \(\alpha+\beta\) is equal to \_\_\_\_ .

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Matrices

For real numbers

α

and

β

, consider the following system of linear equations:

x + y - z = 2, x + 2 y + α z = 1, 2 x - y + z = β

.
If the system has infinite solutions, then

α + β

is equal to ____ .

Answer: 5

Infinitely many solutions require the coefficient determinant to vanish:

112 12 - 1 - 1 α 1 = (2 + α) - (1 - 2 α) - (- 1 - 4) = 3 α + 6 = 0

α = - 2

and the system is

x + y - z = 2

x + 2 y - 2 z = 1

2 x - y + z = β

.
Consistency then needs the third equation to be

p

times the first plus

q

times the second. Matching the coefficients of

x

and

y

gives

p + q = 2

and

p + 2 q = - 1

, so

q = - 3

and

p = 5

, and the coefficient of

z

agrees as well, since

- 5 + 6 = 1

. The right-hand sides must match in the same way, so

β = 5 (2) - 3 (1) = 7

Hence

α + β = - 2 + 7 = 5

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Integer answer