Simultaneous Equations: Equal Values and Substitution Method

Substituting $x=1$ and $y=4$ into both equations shows that both equations are satisfied: $1+4=5$ and $4=3+1$. Therefore, $x=1$ and $y=4$ is the solution to the simultaneous equations.

Substituting $x=3$ and $y=6$ gives $2\times 3+6=12$, which is correct, but $6-3=3$, which is not 8. Therefore, $x=3$ and $y=6$ do not satisfy the simultaneous equations.

Add the two equations together: $(x+y)+(x-y)=5+3$, which simplifies to $2x=8$, so $x=4$. Substituting $x=4$ into $x+y=5$ gives $y=1$. Therefore, the solution is $x=4,y=1$.

Set the two expressions for $y$ equal to each other: $x+3=2x+2$. Solving for $x$ gives $x=1$. Substituting $x=1$ into $y=x+3$ gives $y=4$. Therefore, $x=1$ and $y=4$ is the solution.

Substitute $x=y+2$ into the second equation: $(y+2)+4y=22$. This simplifies to $5y+2=22$, so $5y=20$ and $y=4$. Substituting $y=4$ back gives $x=6$. Therefore, the solution is $x=6,y=4$.

Rearrange the second equation to get $x=y-1$. Substitute this into the first equation: $(y-1)+2y=8$. Simplifying gives $3y-1=8$, so $3y=9$ and $y=3$. Substituting back gives $x=2$. Therefore, the solution is $x=2,y=3$.