Analyzing One-Variable Relationships in Context

The perimeter of a triangle is the sum of its three sides. $△ABC$ has the sides $2x,$ $2x+4,$ and $2x+3.$ Adding these, we can create an algebraic expression that describes the perimeter of the triangle. $$\begin{array}{c}{P}_{△ABC}=2x+(2x+4)+(2x+3)=6x+7\end{array}$$ The triangle $△PQR$ has the sides $3x+1,$ $2x,$ and $2x.$ We can create an expression for the perimeter of this triangle as well. $$\begin{array}{c}{P}_{△PQR}=(3x+1)+2x+2x=7x+1\end{array}$$ Since the perimeters are equal, we can equate the expressions, and solve for $x.$ Knowing that $x=6,$ we can find the side lengths of both triangles by substituting this value into the corresponding expressions.

Expression	$x=6$	Side length
$2x$	$2\cdot 6$	$12$
$2x+3$	$2\cdot 6+3$	$15$
$2x+4$	$2\cdot 6+4$	$16$
$3x+1$	$3\cdot 6+1$	$19$

Therefore, $△ABC$'s side lengths are $12,$ $15$ and $16,$ and the side lengths of $△PQR$ are $12,$ $12$ and $19.$