Two students Filip and Daniel solved the system of equations: $$ \begin{aligned} 2x + 3y &= 5\cr 3x + 2y &= 0 \end{aligned} $$ Each of them solved the system in his own way.
Filip:
(1) He reduced the system of two equations to a single equation by summing them: $$5x + 5y = 5$$ (2) Dividing by $5$, he simplified obtained equation and got: $$ x + y = 1 $$ (3) He concluded that the solution of the given system is every pair of numbers $x$ and $y$ satisfying the equation obtained in step (2). For example, $x=-2$ and $y=3$, etc.
Daniel:
(1) He added $2x+3y$ to the left side and $5$ to the right side of the second equation and obtained the system: $$ \begin{aligned} 2x+3y&=5 \cr 5x+5y&=5 \end{aligned} $$ (2) Dividing the second equation by $5$, he got the equation $x+y=1$, from where: $$ x=1−y $$ (3) He substituted $1-y$ for $x$ to the first equation of the system and obtained: $$ 2(1−y)+3y=5 $$ (4 ) Solving the previous equation he got $y=3$. Then, he substituted $3$ for $y$ back to the equation $x=1−y$ and obtained $x=-2$.
Did both students solve the system of equations correctly, or did one of them make an error?
Filip did not proceed correctly.
Both Filip and Daniel solved the system of equations correctly.
Daniel did not proceed correctly. When we solve a system of equations, we cannot replace an equation in the system with the sum of itself and a different equation of the system.
Neither Filip nor Daniel solved the system of equations correctly.