Given Quadrilateral LMJK is a square. If m∠LMK = (5x + 2y)° and m∠JMK = (11x - y)°, then solve for x and y.

Steps to Solve

1. Identify the angle measures
   Since LMJK is a square, we know that each interior angle (including ∠LMK and ∠JMK) measures $90^\circ$.

2. Set up the equations
   Given:

• $m∠LMK = (5x + 2y)^\circ$
• $m∠JMK = (11x - y)^\circ$
  We can set up the following equations based on the property of a square:
  $$ 5x + 2y = 90 $$
  $$ 11x - y = 90 $$

3. Solve the first equation for y
   Rearranging the first equation gives:
   $$ 2y = 90 - 5x $$
   Thus,
   $$ y = \frac{90 - 5x}{2} $$

4. Substitute y into the second equation
   Substituting the expression for y into the second equation:
   $$ 11x - \left(\frac{90 - 5x}{2}\right) = 90 $$

5. Clear the fraction
   Multiply through by 2 to eliminate the fraction:
   $$ 2(11x) - (90 - 5x) = 180 $$
   This simplifies to:
   $$ 22x - 90 + 5x = 180 $$
   Combining like terms gives:
   $$ 27x - 90 = 180 $$

6. Solve for x
   Now, add 90 to both sides:
   $$ 27x = 270 $$
   Dividing both sides by 27 gives:
   $$ x = 10 $$

7. Substitute x back to find y
   Now that we have $x$, substitute it back into the equation for $y$:
   $$ y = \frac{90 - 5(10)}{2} $$
   Calculating the right side gives:
   $$ y = \frac{90 - 50}{2} = \frac{40}{2} = 20 $$

8. Final solution
   We have found both variables:

• $x = 10$
• $y = 20$