Prove the following statement If a and b are real numbers an

Prove the following statement:
If a and b are real numbers and a= 0, then there exists a unique real number r such that ar + b = 0.

[Hint: there are 2 parts to the proof. Existence of r and Uniqueness of r.
Use a CONSTRUCTIVE proof to show the existence of r; that is, demonstrate what value r must equal.
Use a proof by CONTRADICTION to show the uniqueness of r; that is assume there is more than one r and show all such r’s must equal the same value]

Solution

Existence of R

ar + b = 0

ar = -b

r = -b/a (since a is not equal to zero, hence we can divide the complete equation by a)

since b belongs to R and a belongs to R, therefore -b/a will also belong to R, hence there exists a solution (r=-b/a) that satisfies the equation (ar+b=0)

Uniqueness of r

Let us assume there are two r satisfying this equation i.e. r1 and r2

ar1 + b = 0

ar2 + b = 0

ar1 + b = ar2 + b

(r1=r2)

but our assumption was that r1 and r2 are different, hence our assumption contradicts with the final answer

Hence there is only one r that satisfies this equation