Solve the system

given that,

$x-3y=-25$ ..........(1)

$4x+5y=19$ .........(2)

now, doing $\text{[(1) . 4] and then [(2)-(1)] we get,}$

$4x+5y=19$

$4x-12y=100$

$o r,17y=119$

$o r, y=7$

so, putting the value of y=7 in equation (1) we get, $x=-4$

$s o, x+y=-4+7=3$

$\begin{matrix} \begin{bmatrix}x-3y=-25\\ 4x+5y=19\end{bmatrix}\\\\ x-3y=-25\\ x-3y+3y=-25+3y\\ x=-25+3y\\\\ 4x+5y=19\\ 4\left(-25+3y\right)+5y=19\\ -100+12y+5y=19\\ -100+17y=19\\ -100+17y+100=19+100\\ 17y=119\\ \frac{17y}{17}=\frac{119}{17}\\ y=7\\\\ x=-25+3y\\ x=-25+3(7)\\ x=-25+21\\ x=-4\\\\ x+y\\ =-4+7\\ =\mathbf{3} \end{matrix}$

$x-3y=-25 ....(1)$

$4x+5y=19......(2)$

After multiplying the first equation by 5 and the second equation by 3, we get

$5x-15y = -125$

$12x + 15y = 57$

Now, adding the two equations

$5x+12x = -125+57$

$\Rightarrow 17x = -68$

$\Rightarrow x = -4$

Substituting $x = -4$ in equation $(2)$

$4(-4)+5y=19.$

$\Rightarrow -16+5y=19$

$\Rightarrow 5y = 35$

$\Rightarrow y = 7$

Hence $x + y = -4 + 7 = \boxed{3}$

${\begin{cases} x-3y=-25 \\ 4x+5y=19 \end{cases}}$

Let $(x-3y=-25)$ be line $1)$ and $(4x+5y=19)$ be line $2)$

$1) \ x-3y=-25 \implies x = 3y - 25 \\ 2) \ 4 \cdot (3y-25) +5y = 19 \\ = 17y = 119 \\ = y = \color{#D61F06}7$

Plugging in $y=7$

$2) \ 4x + 5 \cdot 7 = 19 \\ = 4x = 19-35 \\ = x = \color{#3D99F6}-4$

$\color{#3D99F6}x$ $+$ $\color{#D61F06}y$ $=$ $\color{#3D99F6}-4$ $+$ $\color{#D61F06}7$ $= \boxed{3}$

Relevant wiki: System of Linear Equations (Simultaneous Equations)

$x-3y=-25$ $\Longleftrightarrow$ $\color{#D61F06}\boxed{1}$

$4x+5y=19$ $\Longleftrightarrow$ $\color{#D61F06}\boxed{2}$

Solve for $x$ in $\color{#D61F06}\boxed{1}$ in terms of $y$ then substitute in $\color{#D61F06}\boxed{2}$ . We have

$x=-25+3y$

Then,

$4(-25+3y)+5y=19$

$-100+12y+5y=19$

$17y=119$

$y=7$

It follows that

$x=-25+3(7)=-25+21=-4$

Now substitute $x=-4$ and $y=7$ into the two original equations. We have

$x-3y=-25$

$-4-3(7)=-25$

$-25=-25$ (The statement is true.)

$4x+5y=19$

$4(-4)+5(7)=19$

$-16+35=19$

$19=19$ (The statement is true.)

Finally, the desired answer is

$-4+7=\boxed{3}$