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(A) The given system of equations is:$1) 3x + 4y - 5z = -6$$2) 2x + 3y - 4z = -7$$3) 4x - 2y + z = 9$From equation $(3)$,we get $z = 9 - 4x + 2y$.Subs

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$4$

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(A) The given system of equations is:$1) 3x + 4y - 5z = -6$$2) 2x + 3y - 4z = -7$$3) 4x - 2y + z = 9$From equation $(3)$,we get $z = 9 - 4x + 2y$.Substitute $z$ into equation $(1)$:$3x + 4y - 5(9 - 4x + 2y) = -6$$3x + 4y - 45 + 20x - 10y = -6$$23x - 6y = 39$  $(4)$Substitute $z$ into equation $(2)$:$2x + 3y - 4(9 - 4x + 2y) = -7$$2x + 3y - 36 + 16x - 8y = -7$$18x - 5y = 29$  $(5)$Multiply $(4)$ by $5$ and $(5)$ by $6$:$115x - 30y = 195$$108x - 30y = 174$Subtracting these equations: $7x = 21 \implies x = 3$.Substitute $x = 3$ into $(4)$:$23(3) - 6y = 39 \implies 69 - 6y = 39 \implies 6y = 30 \implies y = 5$.Substitute $x = 3, y = 5$ into $(3)$:$z = 9 - 4(3) + 2(5) = 9 - 12 + 10 = 7$.Thus,$(\alpha, \beta, \gamma) = (3, 5, 7)$.Calculate $\alpha + 3\beta - 2\gamma = 3 + 3(5) - 2(7) = 3 + 15 - 14 = 4$.

If $(\alpha, \beta, \gamma)$ is the solution of the system of simultaneous linear equations given by $3x + 4y - 5z = -6$,$2x + 3y - 4z = -7$,and $4x - 2y + z = 9$,then find the value of $\alpha + 3\beta - 2\gamma$.

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