Elimination Method for Solving Simultaneous Equations

Add the two equations together: $(x+y)+(x-y)=8+6$. This eliminates $y$ and simplifies to $2x=14$, so $x=7$. Substituting $x=7$ into either equation gives $7+y=8$, which simplifies to $y=1$. Therefore, the solution is $x=7$ and $y=1$.

Add the two equations together: $(2x+y)+(x-y)=10+2$. This simplifies to $3x=12$, so $x=4$. Substituting $x=4$ into $2x+y=10$ gives $8+y=10$, so $y=2$. Therefore, the solution is $x=4$ and $y=2$.

Add the two equations together: $(x+2y)+(3x-2y)=10+6$. This eliminates $y$ and simplifies to $4x=16$, so $x=4$. Substituting $x=4$ into $x+2y=10$ gives $4+2y=10$, which simplifies to $2y=6$ and $y=3$. Therefore, the solution is $x=4$ and $y=3$.

Subtract the second equation from the first: $(3x+y)-(x+y)=16-4$. This simplifies to $2x=12$, so $x=6$. Substituting $x=6$ into $x+y=4$ gives $6+y=4$, which simplifies to $y=-2$. Therefore, the solution is $x=6$ and $y=-2$.

Subtract the first equation from the second: $(2x+5y)-(2x+y)=36-4$. This eliminates $x$ and simplifies to $4y=32$, so $y=8$. Substituting $y=8$ into $2x+y=4$ gives $2x+8=4$, which simplifies to $2x=-4$ and $x=-2$. Therefore, the solution is $x=-2$ and $y=8$.

First multiply the second equation by 2 to align the coefficients of $y$: $14x-4y=10$. Now add this to the first equation: $(6x+4y)+(14x-4y)=10+10$. This simplifies to $20x=20$, so $x=1$. Substituting $x=1$ into $6x+4y=10$ gives $6+4y=10$, which simplifies to $4y=4$ and $y=1$. Therefore, the solution is $x=1$ and $y=1$.