The value of $y$ varies directly with x, and $y = 40$ when $x = 8$. What is y when $x = 7$?
$y=[?]$
Answer (1)
Establish the direct variation relationship: y = k x .
Calculate the constant of proportionality: k = 8 40 = 5 .
Substitute x = 7 into the equation y = 5 x .
Find the value of y : y = 5 × 7 = 35 , so the final answer is 35 .
Explanation
Understanding the Problem We are given that y varies directly with x , which means that there is a constant k such that y = k x . We are also given that y = 40 when x = 8 . We can use this information to find the constant k .
Finding the Constant of Proportionality Substitute the given values of x and y into the equation y = k x :
40 = k × 8
Calculating k Solve for k by dividing both sides of the equation by 8: k = 8 40 = 5
Finding y when x=7 Now that we have found the constant of proportionality, we can write the equation as y = 5 x . We want to find the value of y when x = 7 .
Calculating y Substitute x = 7 into the equation y = 5 x :
y = 5 × 7 = 35
Examples
Direct variation is a concept that shows up in many real-world situations. For example, the amount you earn at a job where you are paid hourly varies directly with the number of hours you work. If you earn $15 per hour, your total earnings are calculated by multiplying $15 by the number of hours you work. Similarly, the distance a car travels at a constant speed varies directly with the time it travels. If a car travels at 60 miles per hour, the total distance it covers is 60 times the number of hours it travels. Understanding direct variation helps in predicting outcomes in these and other proportional scenarios.
