Solve the System: x + y = 8 and x = 5 - y Using Substitution

Note that in the current system of equations, one of the variables is isolated alone on the left side of the equation:

$\begin{cases} \underline{x}+y=8 \\ \bm{x=\underline{5-y}} \\ \end{cases}$

Therefore, we can apply the substitution method and substitute the entire expression that x equals in the second equation in place of x in the first equation (marked with an underline in both equations above) Hence we obtain one equation with one variable:

$\underline{ 5-y}+y=8$

Highlight the equation in which the variable we substituted is isolated in order to return to it later.

From here - we'll proceed to solve the single-variable equation that we obtained.

First- combine like terms on the left side of the resulting equation:

$5-y+y=8 \\ 5=8$$$

Note that y cancelled out in the current equation and we obtained a false statement, as shown below:

$5\neq8$ meaning-

We obtained a false statement regardless of the variables' values,

We can conclude from here that the system of equations has no solution, given that no matter which values we substitute for the variables - we won't obtain a true statement in both equations together.

Therefore the correct answer is answer D.