To solve the system of linear differential equations using elimination, we start with the given system:

$x' = 3x - y$ $y' = 3y - 4x$

Our goal is to eliminate $x$ and solve for $y$. Let's proceed step-by-step:

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Step 1: Differentiate $x' = 3x - y$ with respect to $t$

We want to differentiate both sides to get an expression involving $x''$ and eliminate $x$:

$x'' = 3x' - y'$

Using the second equation $y' = 3y - 4x$, substitute $y'$ into the above equation:

$x'' = 3x' - (3y - 4x)$

\tag{1}$$ --- ### Step 2: Substitute $$x' = 3x - y$$ into the expression We substitute $$y = 3x - x'$$ to solve .