SOLUTION: Emily counts the triangles and parallelograms in an art piece and determines that there are altogether 42 triangles and parallelograms. If there are 150 total sides, how many trian

Based on the context of the problem, I assume that these triangles and parallelograms are separate shapes / areas / figures

that are located at totally different parts of the art piece and do not intersect each other and do not have common

sides and/or vertices.


Then from the condition, you have this system of equations for the number of triangles T and the number of parallelograms  P


     T +  P =  42     (1)    (counting shapes altogether)

    3T + 4P = 150     (2)    (counting sides)


To solve the system, express  T = 42 - P  from equation (1) and substitute it into equation (2), replacing T there


    3*(42-P) + 4Q = 150.


Simplify step by step


    126 - 3P + 4P = 150

     P = 150 - 126 = 24.


ANSWER.  There are 24 parallelograms and 42-24 = 18 triangles in the art piece.


CHECK.   4*24 + 18*3 = 150,  the total number of sides.   ! Precisely correct !