If there is inverse variation and [tex]y=6[/tex] when [tex]x=-2[/tex], find [tex]y[/tex] when [tex]x=-3[/tex].
A. 1
B. -4
C. 4
D. -1
Answer (2)
The problem states that x and y vary inversely, meaning x y = k for some constant k .
First, find the constant of variation k using the given values x = − 2 and y = 6 , which gives k = − 12 .
Then, substitute the new value x = − 3 into the equation x y = − 12 .
Solve for y by dividing both sides by − 3 , resulting in y = 4 .
The final answer is 4 .
Explanation
Understanding Inverse Variation We are given that x and y vary inversely. This means that their product is a constant, i.e., x y = k for some constant k . We are given that y = 6 when x = − 2 . We need to find the value of y when x = − 3 .
Finding the Constant of Variation First, we use the given information to find the constant of variation k . We have x = − 2 and y = 6 , so we substitute these values into the equation x y = k :
( − 2 ) ( 6 ) = k k = − 12
Substituting the New Value of x Now we know that the relationship between x and y is given by the equation x y = − 12 . We want to find the value of y when x = − 3 . We substitute x = − 3 into the equation: ( − 3 ) y = − 12
Solving for y To solve for y , we divide both sides of the equation by − 3 :
y = − 3 − 12 y = 4
Final Answer Therefore, when x = − 3 , y = 4 .
Examples
Inverse variation is a concept that appears in many real-world scenarios. For example, the time it takes to complete a journey is inversely proportional to the speed at which you travel. If you double your speed, you halve the time it takes to reach your destination, assuming the distance remains constant. Similarly, in physics, the pressure of a gas is inversely proportional to its volume at a constant temperature. Understanding inverse variation helps in making predictions and optimizing various processes in everyday life.
The value of y when x = − 3 is determined to be 4, so the answer is option C. To find this, we used the inverse variation relationship to calculate the constant and then solve for y .
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