Discrete Math ProofsAre these steps for finding the solutions of x + 3 = 3...

anudoneddbv

anudoneddbv

Answered

2022-07-17

Discrete Math Proofs
Are these steps for finding the solutions of x + 3 = 3 xcorrect?
1. x + 3 = 3 x is given;
2. x + 3 = x 2 6 x + 9, obtained by squaring both sides of (1);
3. 0 = x 2 7 x + 6, obtained by subtracting x + 3 from both sides of (2);
4. 0 = ( x 1 ) ( x 6 ), obtained by factoring the right-hand side of (3);
5. x = 1 or x = 6, which follows from (4) because a b = 0 implies that a = 0 or b = 0.

Answer & Explanation

Clarissa Adkins

Clarissa Adkins

Expert

2022-07-18Added 16 answers

Step 1
Those steps are correct, but there's one final step required:
6. When x = 1, the left-hand side is 2 and the right-hand side is 2; when x = 6, the left-hand side is 3 and the right-hand side is -3. So only the first "solution" is a solution. (Er. When I did this the first time, I wrongly thought that the RHS in the x = 6 case was 3.)
To explain step 5: we want to find x, given that ( x 1 ) ( x 6 ) = 0. Let a = x 1 and b = x 6; then since a b = 0, we have a = 0 or b = 0. So x 1 = 0 or x 6 = 0; so x = 1 or x = 6.
Step 2
Why is it the case that if a b = 0 then a = 0 or b = 0? This is precisely the statement that the product of nonzero quantities is nonzero (by taking the contrapositive), which you might find more intuitive; but I see there is another answer which gives algebraic manipulations to prove it.
Damien Horton

Damien Horton

Expert

2022-07-19Added 5 answers

Step 1
Yes, all the steps are correct. You only need to check those value of x if it satisfies the original equation or not.
Assume that a b = 0. We want to show that either a = 0 or b = 0.
Step 2
If a = 0 then we are done. Suppose that a 0. Then a 1 R . Thus, using field properties of R we get b = 1 b = ( a 1 a ) b = a 1 ( a b ) = a 1 0 = 0.

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