Full solution

Q. Solve the system.

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\begin{cases} x+3y-z&=-6 \ -2x-y+z&=10 \ x-y+3z&=2 \end{cases}

Write Equations: Write down the system of equations. $\newline$ The system of equations is: $\newline$ $x + 3y - z = -6$ $\newline$ $-2x - y + z = 10$ $\newline$ $x - y + 3z = 2$
Elimination Method: Use the elimination method to eliminate one variable. $\newline$ We can start by eliminating the variable $z$ . To do this, we can add the first and second equations to eliminate $z$ . $\newline$ $(x + 3y - z) + (-2x - y + z) = -6 + 10$ $\newline$ This simplifies to: $\newline$ $-x + 2y = 4$
Solve for $x$ and $y$ : Now we have a new system of two equations with two variables. $\newline$ The new system is: $\newline$ $-x + 2y = 4$ $\newline$ $x - y + 3z = 2$
Equations with y and z: Solve the new system for x and y. $\newline$ We can solve for x from the first equation of the new system: $\newline$ $-x + 2y = 4$ $\newline$ $x = 2y - 4$ $\newline$ Now we substitute x in the second equation: $\newline$ $(2y - 4) - y + 3z = 2$ $\newline$ $y - 4 + 3z = 2$ $\newline$ $y + 3z = 6$
Solve for y and z: We now have two equations with y and z. $\newline$ The equations are: $\newline$ $y + 3z = 6$ $\newline$ $-2x - y + z = 10$ $\newline$ We can substitute x from step $4$ into the second equation: $\newline$ $-2(2y - 4) - y + z = 10$ $\newline$ $-4y + 8 - y + z = 10$ $\newline$ $-5y + z = 2$
Find $z$ : Solve the system of equations with $y$ and $z$ . We have two equations: $y + 3z = 6$ $-5y + z = 2$ We can multiply the first equation by $5$ to eliminate $y$ : $5(y + 3z) = 5(6)$ $5y + 15z = 30$ Now we add this to the second equation: $5y + 15z + (-5y + z) = 30 + 2$ This simplifies to: $16z = 32$ $z = \frac{32}{16}$ $z = 2$
Find $y$ : Substitute $z$ back into one of the equations to find $y$ . We can use the equation $y + 3z = 6$ : $y + 3(2) = 6$ $y + 6 = 6$ $y = 6 - 6$ $y = 0$
Find $x$ : Substitute $y$ back into the equation from step $4$ to find $x$ . We use the equation $x = 2y - 4$ : $x = 2(0) - 4$ $x = -4$
Final Solution: Write down the solution to the system of equations. $\newline$ The solution is $x = -4$ , $y = 0$ , $z = 2$ .