Why Are Two Coefficients of t in the t-Distribution Positive?

[s3a]

I'm attaching the question, answer, and my work. My work is almost 100% correct except that there are two coefficients of t that are supposed to be positive instead of negative and I marked those with red writing and would appreciate an explanation as to why this is the case.

Thanks in advance!

 

[oli4]

You have a sign error because you forgot to transform the equation in its (f(x,y,z)=0 form
that is, from the first formula you must substract z
that would give you the function F(x,y,z)=x²+2xy+5y²+x+3y+1-z
(notice that z changes sign which is what did not happen in your solution)
if you continue from there, you will get to a solution identical to the known answer (except maybe t will have a minus sign, but everywhere, so it doesn't change anything)

Cheers...

 

[Mark44]

I'm attaching the question, answer, and my work. My work is almost 100% correct except that there are two coefficients of t that are supposed to be positive instead of negative and I marked those with red writing and would appreciate an explanation as to why this is the case.

Thanks in advance!

For your tangent plane you have z = -13x - 43y - 112. You can also write this in standard form as 13x + 43y + z = -12. From this form, a normal vector can be obtained by inspection: <13, 43, 1>.

 

[s3a]

Thanks guys, I get it now! :)