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. $y=\frac{?}{2}$
D. $y=\frac{2}{(48)(6)}$
Answer (2)
Establish the direct variation relationship: y = k x .
Use the given values x = 6 and y = 48 to find the constant of proportionality: k = 6 48 = 8 .
Substitute k back into the equation: y = 8 x .
Replace x with Z to find the expression for y when x = Z : y = 8 Z = 6 48 Z . The final answer is y = 6 48 Z .
Explanation
Understanding the Problem 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 when x = 6 , y = 48 . We want to find an expression for y when x = Z .
Setting up the Equation Since y varies directly as x , we can write the relationship as y = k x , where k is the constant of proportionality.
Finding the Constant of Proportionality We are given that y = 48 when x = 6 . We can use this information to find the constant of proportionality k . Substitute these values into the equation y = k x to get 48 = k ( 6 )
Calculating k To solve for k , we divide both sides of the equation by 6: k = 6 48 = 8
Direct Variation Equation Now that we have the constant of proportionality, the direct variation equation is y = 8 x .
Finding y when x = Z To find the value of y when x = Z , substitute Z for x in the equation y = 8 x . This gives y = 8 Z We can also write 8 as 6 48 , so y = 6 48 Z
Final Answer The expression that can be used to find the value of y when x is Z is y = 6 48 Z .
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 48 miles in 6 hours, then the distance you travel in Z hours at the same speed is (48/6)*Z. This concept is also applicable in calculating the cost of items when the price per item is constant, or in determining the amount of ingredients needed when scaling up a recipe.
The expression that can be used to find the value of y when x is 2 is option A, y = 6 48 ( 2 ) . This expression shows the direct variation relationship derived from the given information. By calculating the value with this expression, we can find that y = 16 .
;
