I want to calculate possible T natural numbers for which the following two equations are true:
$$T=8x+7 $$ $$T=6y+5$$
Wolfram Alpha gives me the function:
Integer solution:
$T = 24 n + 23$, where $x = 3 n + 2$ and $y = 4 n + 3$, $\forall n\in\mathbb{Z}$
I have not done real math since High School, decades ago. What steps does one follow to get to that function?
P.S.: Symbolab returns something that I cannot see how it maps to WA’s answer. How are both results related?
Equating the two expressions for $T$, and simplifying: $$8x+7=6y+5\\ 8x+2=6y\\ 4x+1=3y$$ The RHS is a multiple of $3$, so the left hand side needs to be a multiple of $3$ as well. If you just try some values $0,1,2,3..$ for $x$, you will find that the only ones that make $4x+1$ a multple of $3$ are $2,5,8,11,...$. It is easy to see that this pattern continues, as adding a multiple of $3$ to $x$ will not affect whether the left hand side is divisible by $3$. So we get that $x=3n+2$ for some integer $n$.
A more formal way to solve this is by modular arithmetic, where all multiples of $3$ are ignored: $$4x+1\equiv3y \pmod 3\\ 3x+x+1\equiv3y \pmod 3\\ x+1\equiv0 \pmod 3\\ x\equiv-1 \pmod 3\\ x\equiv-1+3=2 \pmod 3\\ x=3n+2$$
Subtituting this into $4x+1=3y$ we get $$4(3n+2)+1=3y\\ 12n+8+1=3y\\ 12n+9=3y\\ 4n+3=y$$
So we get that $y=4n+3$ and $x=3n+2$. Substituting either of these into one of the expressions for $T$ gives that $T=24n+23$.
Symbolab gives a different answer because it does not know that you want $x,y$ to be integers. It expresses $T$ and $y$ in terms of $x$, but the expression for $y$ is as a fraction. To make that fraction simplify and get an integer $y$, you would still need to restrict the values of $x$ to $x=3n+2$ for integer $n$.