Question

independent practice for exercises 10-12 see example 1, 13-14 2, 15-16 3, 17 4. extra practice skill practice p. s13 application practice p. s32. 15. the value of y varies directly with x, and y = 8 when x = -32. find y when x = 64. 16. the value of y varies directly with x, and y = 1/2 when x = 3. find y when x = 1. 17. while on his way to school, norman saw that the cost of gasoline was $2.50 per gallon. write a direct variation equation to describe the cost y of x gallons of gas. then graph. tell whether each relationship is a direct variation. explain your answer. 18. the equation -15x + 4y = 0 relates the length of a videotape in inches x to its approximate playing time in seconds y. 19. the equation y - 2.00x = 2.50 relates the cost y of a taxicab ride to distance x of the cab ride in miles. each ordered pair is a solution of a direct variation. write the equation of direct variation. then graph your equation and show that the slope of the line is equal to the constant of variation. 20. (2, 10) 21. (-3, 9) 22. (8, 2) 23. (1.5, 6) 24. (7, 21) 25. (1, 2) 26. (2, -16) 27. (1/7, 1) 28. (-2, 9) 29. (9, -2) 30. (4, 6) 31. (3, 4) 32. (5, 1) 33. (1, -6) 34. (-1, 1/2) 35. (7, 2) 36. astronomy weight varies directly with gravity. a mars lander weighed 767 pounds on earth but only 291 pounds on mars. its accompanying mars rover weighed 155 pounds on mars. how much did it weigh on earth? round your answer to the nearest pound. 37. environment mischa bought an energy - efficient washing machine. she will save about 15 gallons of water per wash load. a. write an equation of direct variation to describe how many gallons of water y mischa saves for x loads of laundry she washes. b. graph your direct variation from part a. is every point on the graph a solution in this situation? why or why not? c. if mischa does 2 loads of laundry per week, how many gallons of water will she have saved at the end of a year? 38. critical thinking if you double an x - value in a direct variation, will the corresponding y - value double? explain. 39. write about it in a direct variation y = kx, k is sometimes called the “constant of proportionality.” how are proportions related to direct variations? 40. this problem will prepare you for the multi - step test prep on page 342. rhea exercised on a treadmill at the gym. when she was finished, the display showed that she had walked at an average speed of 3 miles per hour. a. write an equation that gives the number of miles y that rhea would cover in x hours if she walked at this speed. b. explain why this is a direct variation and find the value of k. what does this value represent in rhea’s situation?

Explanation:

Let's solve problem 15: "The value of \( y \) varies directly with \( x \), and \( y = 8 \) when \( x = -32 \). Find \( y \) when \( x = 64 \)."

Step 1: Recall the direct variation formula

The formula for direct variation is \( y = kx \), where \( k \) is the constant of variation.

Step 2: Find the constant \( k \)

We know that \( y = 8 \) when \( x = -32 \). Substitute these values into the formula:
\[
8 = k(-32)
\]
Solve for \( k \):
\[
k = \frac{8}{-32} = -\frac{1}{4}
\]

Step 3: Write the direct variation equation

Now that we have \( k = -\frac{1}{4} \), the equation is \( y = -\frac{1}{4}x \).

Step 4: Find \( y \) when \( x = 64 \)

Substitute \( x = 64 \) into the equation:
\[
y = -\frac{1}{4}(64) = -16
\]

Answer:

When \( x = 64 \), \( y = -16 \).