What is the constant of variation, [tex]$k$[/tex], of the direct variation, [tex]$y=k x$[/tex], through [tex]$(-3,2)$[/tex]?
A. [tex]$k=-\frac{3}{2}$[/tex]
B. [tex]$k=\frac{2}{3}$[/tex]
C. [tex]$k=\frac{2}{3}$[/tex]
D. [tex]$k=\frac{3}{2}$[/tex]
Answer (2)
Substitute the given point ( − 3 , 2 ) into the direct variation equation y = k x .
Solve for k : 2 = k ( − 3 ) .
Divide both sides by − 3 to find k = − 3 2 .
The constant of variation is − 3 2 .
Explanation
Understanding the Problem We are given a direct variation equation y = k x , where k is the constant of variation. We are also given that the line passes through the point ( − 3 , 2 ) . This means that when x = − 3 , y = 2 . Our goal is to find the value of k .
Solving for k To find the constant of variation k , we substitute the given point ( − 3 , 2 ) into the equation y = k x . This gives us: 2 = k ( − 3 ) Now, we solve for k by dividing both sides of the equation by − 3 : k = − 3 2 k = − 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. Similarly, the amount you earn if you are paid hourly varies directly with the number of hours you work. If you earn 15 p er h o u r , yo u re a r nin g s e a re g i v e nb y e = 15h , w h ere h$ is the number of hours you work.
The constant of variation k for the direct variation equation through the point ( − 3 , 2 ) is calculated to be k = − 3 2 , which is not listed among the given options. It's derived by substituting the point into the equation y = k x and solving for k . None of the answer choices provided are correct.
;
