Final answer:
In direct variation where x = 30y, solutions for the conditions xy = 750, x + y = 155, and ratios x = 30y or x/y = 30 affirm the direct relationship, yielding x = 150 and y = 5 whenever these conditions are applied.
Step-by-step explanation:
When x varies directly as y, there exists a constant k such that x = ky. Given that x = 150 when y = 5, we find the constant using the equation 150 = k × 5. Solving for k, we get k = 30. This constant allows us to solve for other relations involving x and y.
- a) To find x when the product xy = 750, we substitute k into the direct variation equation x = 30y and set it equal to 750: 30y × y = 750. Solving for y, we find that y = 5 and x = 150.
- b) When x = 30y, this is simply the direct variation equation we've already established, meaning this relationship is always true for any values of x and y that maintain the variation.
- c) For x + y = 155, substitute x = 30y into the equation: 30y + y = 155. This simplifies to 31y = 155. Dividing both sides by 31, we find that y = 5 and therefore x = 150.
- d) The ratio x/y = 30 reflects the constant of variation; thus, this equation is also always true for any set of values x and y that maintain the direct variation.