Question

Give an example of a linear system of three equations with three unknowns that has the general solution with one parameter.

Answer 1

Ans. Example:

system of three equations in three unknowns.
x1+ x2 + x3 = 4 (1)
x1+ 2x2 + 3x3 = 9 (2)
2x1+ 3x2 + x3 = 7 (3)
We can reduce the system down to two equations in two unknowns by using the first equation to solve for
x1 in terms of x2 and x3
x1 = 4 − x2 − x3 equation-(1’)
and substituting this solution into the remaining two equations
(2) (4 − x2 − x3) + 2x2+3x3 = 9 =⇒ x2+2x3 = 5
(3) 2(4 − x2 − x3) + 3x2+ x3 = 7 =⇒ x2− x3 = −1
We now have two equations in two unknowns, x2 and x3. We can solve the first of these two equations for
x2 in terms of x3
x2 = 5 − 2x3 equation -(2’)

and substitute the result into the final equation
(5 − 2x3) − x3 = −1 =⇒ −3x3 = −6
to get down to one equation in the one unknown x3. We can trivially solve it for x3
x3 = 2
and substitute the result into (2’) to give x2
x2 = 5 − 2x3 = 5 − 2 × 2 = 1
and finally substitute the now known values of x2 and x3 into (1’) to determine x1
x1 = 4 − x2 − x3 = 4 − 1 − 2 = 1