Solve the given simultaneous equation to find the value of y:
[tex]\begin{gathered} \begin{cases}4x-2y+3z=1 \\ 8x-3y+5z=4 \\ 7x-2y+4z=5\end{cases} \\ \text{Multiply the first equation by 2 to get: } \\ 8x-4y+6z=2 \end{gathered}[/tex]
Subtract the new equation from the second equation:
[tex]\begin{gathered} \begin{cases}8x-3y+5z=4 \\ 8x-4y+6z=2\end{cases} \\ \text{Subtract to get:} \\ -3y-(-4y)+5z-6z=4-2_{} \\ \Rightarrow-3y+4y-z=2 \\ \Rightarrow y-z=2 \end{gathered}[/tex]
Multiply the first equation by 7 and the last equation by 4, then subtract:
[tex]\begin{gathered} \begin{cases}28x-14y+21z=7 \\ 28x-8y+16z=20\end{cases} \\ \text{Subtract the second equation from the first:} \\ \Rightarrow-14y-(-8y)+21z-16z=7-20 \\ \Rightarrow-6y+5z=-13 \end{gathered}[/tex]
Combine this equation with the first one derived, y-z=2.
[tex]\begin{gathered} \begin{cases}-6y+5z=-13 \\ y-z=2\end{cases} \\ \text{Multiply the second equation by 5:} \\ \begin{cases}-6y+5z=-13 \\ 5y-5z=10\end{cases} \\ Add\text{ the equations:} \\ \Rightarrow-6y+5y+5z-5z=-13+10 \\ \Rightarrow-y=-3 \\ \text{Divide both sides by -1:} \\ \Rightarrow y=3 \end{gathered}[/tex]
Hence, the value of y is 3.
The correct option is A.
