If $y$ varies directly as $x$, and $y$ is 48 when $x$ is 6, which expression can be used to find the value of $y$ when $x$ is $2$?
A. $y=\frac{48}{6}(2)$
B. $y=\frac{6}{48}(2)$
C. $y=\frac{(48)(6)}{2}$
D. $y=\frac{2}{(48)(6)}$
Answer (2)
Establish the direct variation relationship: y = k x .
Use given values to find the constant of proportionality: k = 6 48 = 8 .
Substitute x = 2 into the equation: y = 8 ( 2 ) .
The expression to find the value of y when x is 2 is: y = 6 48 ( 2 )
Explanation
Understanding Direct Variation We are given that y varies directly as x . This means that there is a constant k such that y = k x . We are also given that y = 48 when x = 6 . We can use this information to find the constant of proportionality k .
Finding the Constant of Proportionality Substitute y = 48 and x = 6 into the equation y = k x to find k :
48 = k ( 6 )
Solving for k Divide both sides of the equation by 6 to solve for k :
k = 6 48 = 8
Writing the Direct Variation Equation Now that we have found k , we can write the direct variation equation as y = 8 x . We want to find the value of y when x = 2 .
Finding y when x = 2 Substitute x = 2 into the equation y = 8 x :
y = 8 ( 2 ) = 16
The Expression The expression to find the value of y when x is 2 is y = 6 48 ( 2 ) .
Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the distance you travel at a constant speed varies directly with the time you spend traveling. If you travel 100 miles in 2 hours, the relationship between distance and time can be expressed as d = k t , where d is the distance, t is the time, and k is the constant speed. In this case, 100 = k ( 2 ) , so k = 50 miles per hour. This means that for every hour you travel, you cover 50 miles. Understanding direct variation helps in predicting outcomes based on constant relationships.
The expression to find the value of y when x = 2 is y = 6 48 ( 2 ) , which corresponds to answer choice A. This is derived from understanding that y varies directly as x and finding the constant of proportionality. We used the initial conditions to confirm this relationship.
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