Peter solved the given system of equations $$\begin{aligned} 2(y+x)-(y+1)&=x+2\cr 3(x-1)+2(y+x)&=3x+y+2 \end{aligned}$$ in the following way:
(1) He removed the parentheses: $$\begin{aligned} 2y+x−y+1&=x+2 \cr 3x-1+2y+x&=3x+y+2 \end{aligned}$$
(2) Then he simplified the equations by adding $(-x)$ to both sides of the first equation and adding $(-3x−y)$ to both sides of the second equation, resulting in: $$\begin{aligned} y+1&=2 \cr -1+y+x&=2 \end{aligned}$$
(3) From the equations above, he easily determined that $y=1$ and $x=2$.
(4) Finally, he checked the solution: $$ L_1=2(1+2)-(1+1)=4,~R_1=2+2=4 \Rightarrow L_1=R_1 $$ $$ L_2=3(2-1)+2(1+2)=9,~R_2=3\cdot 2+1+2=9 \Rightarrow L_2=R_2 $$ The teacher gave Peter for his solution an insufficient grade. Petr asked his classmates for comments. Which one is correct?
Rebeca claims that Peter made an error in step (1). He used the distributive property incorrectly.
Karel is convinced that the teacher is wrong because he did not notice that the check turned out well.
Bill thinks that Peter made an error in step (4) when doing the check.
Alan is convinced that Peter made an error in step (3). From the above equations, it is clear that the system has no solution. The right sides of the equations are the same, but the left sides are different.
To remove parentheses, it is necessary to distribute the number in front of the parentheses to each term inside of the parentheses.