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The quantity z z varies directly with y y and inversely with x x . When y=12 y = -12 and x=4 x = -4 , z=36 z = -36 . What is the equation of variation? Write your answer in the form A A or AB \frac{A}{B} , where A A and B B are constants or variable expressions. Remember to include the value of k k , the constant of variation, in exact form.\newline  z\ z =______

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Q. The quantity z z varies directly with y y and inversely with x x . When y=12 y = -12 and x=4 x = -4 , z=36 z = -36 . What is the equation of variation? Write your answer in the form A A or AB \frac{A}{B} , where A A and B B are constants or variable expressions. Remember to include the value of k k , the constant of variation, in exact form.\newline  z\ z =______
  1. Write Variation Equation: Since zz varies directly with yy and inversely with xx, the equation of variation can be written as z=kyxz = k\cdot\frac{y}{x}, where kk is the constant of variation.
  2. Substitute Given Values: We are given that when y=12y = -12 and x=4x = -4, z=36z = -36. We can substitute these values into the equation to find kk: 36=k(12/4)-36 = k\cdot(-12/-4).
  3. Simplify Equation: Simplifying the equation, we get 36=k×3-36 = k \times 3.
  4. Solve for k: Now, we solve for kk by dividing both sides by 33: k=363k = -\frac{36}{3}.
  5. Find Value of k: After dividing, we find that k=12k = -12.
  6. Final Variation Equation: With the value of kk found, we can write the equation of variation as z=12(yx)z = -12\cdot\left(\frac{y}{x}\right).

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