Answer
$x=-2$
The side lengths are $6$, $8$, and $5$.
Work Step by Step
We are given that the perimeter of the triangle is $19$, and the side lengths are $3x+12$, $-4x$, and $1-2x$. Thus, we can add up the side lengths to find what the perimeter is in terms of $x$.
$(3x+12)+(-4x)+(1-2x)=13-3x$
Since $13-3x$ is equal to the perimeter, and the perimeter is also $19$, this must also mean that $13-3x=19$.
To solve this equation, we can first subtract $13$ on both sides to isolate the variable $x$: $-3x=6$.
Then, divide by $-3$ on both sides to remove the coefficient: $x=-2$.
Thus, $x=-2$.
To find each of the side lengths, plug $-2$ back into each side length expression and simplify.
$3(-2)+12=-6+12=6$
$-4(-2)=8$
$1-2(-2)=1+4=5$
