Question

If a linear system has four equations and seven variables, then it must have infinitely many solutions. This statement is false, could you explain why and give an example?

Answer 1

Answer:

False. See the explanation below.

Step-by-step explanation:

We need to proof if the following statement "If a linear system has four equations and seven variables, then it must have infinitely many solutions." is false.

And the best way to proof that is false is with a counterexample.

Let's assume that we have seven random variables given by $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ and we have the following four equations given by the following system:

$a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 =1$   (1)

$a_1 +a_2= 0$    (2)

$a_3 +a_4 +a_5 =1$   (3)

$a_6 +a_7 =1$   (4)

As we can see we have system and is inconsistent since equation (1) is not satisfied by equation (2) ,(3) and (4) if we add those equations we got:

$a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 = 0+1+1= 2 \neq 1$

So then we can have a system of 7 variables and 4 equations inconsistent and with not infinitely solutions for this reason the statement is false.