Question

problem solving the sides of \\(\parallelogram mnpq\\) are represented by the expressions below. find the perimeter of \\(\parallelogram mnpq\\).\\(mq = -2x + 37\\) \\(qp = y + 14\\)\\(np = x - 5\\) \\(mn = 4y + 5\\)\\(\text{perimeter: } \square \text{ units}\\)

Explanation:

Step1: Set equal opposite sides

In a parallelogram, opposite sides are equal, so:
$MQ = NP$: $\quad -2x + 37 = x - 5$
$MN = QP$: $\quad 4y + 5 = y + 14$

Step2: Solve for x

Isolate x terms and solve:
$-2x - x = -5 - 37$
$-3x = -42$
$x = \frac{-42}{-3} = 14$

Step3: Solve for y

Isolate y terms and solve:
$4y - y = 14 - 5$
$3y = 9$
$y = \frac{9}{3} = 3$

Step4: Calculate side lengths

Substitute x=14 into $MQ$:
$MQ = -2(14) + 37 = -28 + 37 = 9$
Substitute y=3 into $MN$:
$MN = 4(3) + 5 = 12 + 5 = 17$

Step5: Compute perimeter

Perimeter of parallelogram: $2\times(\text{side 1} + \text{side 2})$
$2\times(9 + 17) = 2\times26 = 52$

Answer:

52 units