The value of $y$ varies directly with $x$, and $y=-16$ when $x=4$. What is y when $x =12$?
Answer (2)
Determine the constant of proportionality k using the given values x = 4 and y = − 16 in the equation y = k x , which gives k = − 4 .
Substitute x = 12 and k = − 4 into the equation y = k x .
Calculate y = ( − 4 ) ( 12 ) .
The value of y when x = 12 is − 48 .
Explanation
Find the constant of proportionality We are given that y varies directly with x , which means that y = k x for some constant k . We are also given that y = − 16 when x = 4 . We can use this information to find the value of k .
Calculate k Substitute y = − 16 and x = 4 into the equation y = k x :
− 16 = k ( 4 ) To solve for k , divide both sides by 4: k = 4 − 16 = − 4
Calculate y when x=12 Now that we have the value of k , we can find the value of y when x = 12 . Substitute k = − 4 and x = 12 into the equation y = k x :
y = ( − 4 ) ( 12 ) = − 48
Final Answer Therefore, when x = 12 , y = − 48 .
Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the amount you earn at a job that pays an hourly wage varies directly with the number of hours you work. If you earn $15 per hour, your total earnings are directly proportional to the hours you put in. Similarly, in physics, the distance an object travels at a constant speed varies directly with the time it travels. Understanding direct variation helps in predicting outcomes and making informed decisions in various practical situations.
To find the value of y when x = 12 , we first determined the constant of proportionality k = − 4 . Using this, we calculated y to be − 48 when x = 12. T h u s , t h e an s w er i s y = -48$.
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