What is the constant of variation, $k$, of the direct variation, $y=k x$, through $(-3,2)$?
A. $k=-\frac{3}{2}$
B. $k=-\frac{2}{3}$
C. $k=\frac{2}{3}$
D. $k=\frac{3}{2}
Answer (2)
Substitute the given point ( − 3 , 2 ) into the direct variation equation y = k x .
Obtain the equation 2 = k ( − 3 ) .
Solve for k by dividing both sides by − 3 .
The constant of variation is − 3 2 .
Explanation
Understanding the Problem We are given a direct variation equation y = k x and a point ( − 3 , 2 ) that lies on the line. Our goal is to find the constant of variation, k .
Substituting the Point To find k , we substitute the coordinates of the given point into the equation. So, we have 2 = k × ( − 3 ) .
Solving for k Now, we solve for k by dividing both sides of the equation by − 3 : k = − 3 2 = − 3 2
Final Answer Therefore, the constant of variation is k = − 3 2 .
Examples
Direct variation is a relationship between two variables where one is a constant multiple of the other. For example, the distance you travel at a constant speed varies directly with the time you spend traveling. If you travel at a speed of 60 miles per hour, the distance d you travel is given by d = 60 t , where t is the time in hours. This means that for every hour you travel, the distance increases by 60 miles. Understanding direct variation helps in many real-world scenarios, such as calculating travel distances, determining the cost of items based on quantity, and understanding proportional relationships in science and engineering.
The constant of variation for the direct variation equation y = k x through the point ( − 3 , 2 ) is found by substituting the coordinates into the equation. Solving gives k = − 3 2 . Therefore, the correct answer is option B.
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