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?

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 [tex]a_1, a_2, a_3, a_4, a_5, a_6, a_7[/tex] and we have the following four equations given by the following system:

[tex] a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 =1[/tex] (1)

[tex] a_1 +a_2= 0[/tex] (2)

[tex] a_3 +a_4 +a_5 =1[/tex] (3)

[tex] a_6 +a_7 =1[/tex] (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:

[tex] a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 = 0+1+1= 2 \neq 1[/tex]

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.