Question 205479
Suppose you need to construct a right triangle in which the shortest side is eight feet less than the largest side and the third side is seven feet more than the shortest side.
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{{{Smallest = Longest - 8}}}
{{{Middle = Smallest + 7}}}
{{{Smallest^2 + Middle^2 = Longest^2}}}

{{{system(S=L-8,M=S+7,S^2+M^2=L^2)}}}

Substitute {{{L-8}}} for {{{S}}} in {{{M=S+7}}}

{{{M=S+7}}}
{{{M=(L-8)+7}}}
{{{M=L-8+7}}}
{{{M=L-1}}}

Substitute {{{L-8}}} for {{{S}}} and {{{L-1}}} for M in {{{S^2+M^2=L^2}}}

{{{S^2+M^2=L^2}}}
{{{(L-8)^2+(L-1)^2=L^2}}}
{{{(L-8)(L-8)+(L-1)(L-1)=L^2}}}
{{{(L^2-8L-8L+64)+(L^2-L-L+1)=L^2}}}
{{{(L^2-16L+64)+(L^2-2L+1)=L^2}}}
{{{L^2-16L+64+L^2-2L+1=L^2}}}
{{{2L^2-18L+65=L^2}}}
{{{L^2-18L+65=0}}}
{{{(L-13)(L-5)=0}}}
{{{L-13=0}}}   {{{L-5=0}}}
   {{{L=13}}}    {{{L=5}}}

So the longest side is 13 ft or 5 ft
But we must check:

Substituting {{{L=13}}}   

{{{Smallest = Longest - 8 = 13-8=5}}}
{{{Middle = Smallest + 7=5+7=12}}}

That's one solution. Longest = 13 ft, Middle = 12 ft, Longest = 13 ft.

Substituting {{{L=5}}}   

{{{Smallest = Longest - 8 = 5-8=-3}}}
{{{Middle = Smallest + 7=-3+7=4}}}

That would require a triangle to have a negative side,
so we discard that as a solution.

Edwin</pre>