If there is direct variation and [tex]y=-5[/tex] when [tex]x=2[/tex], find [tex]y[/tex] when [tex]x=-6[/tex].
A. 60
B. -3
C. 15
D. 0.60
Answer (1)
Establish the direct variation relationship: y = k x .
Determine the constant of variation: k = 2 − 5 .
Substitute x = − 6 into the equation: y = 2 − 5 ( − 6 ) .
Calculate the value of y : 15 .
Explanation
Finding the Constant of Variation We are given that y varies directly with x , which means that there exists a constant k such that y = k x . We are also given that when x = 2 , y = − 5 . We can use this information to find the constant of variation k .
Determining the Equation Substitute x = 2 and y = − 5 into the equation y = k x :
− 5 = k ( 2 ) To solve for k , divide both sides by 2: k = 2 − 5 So, the direct variation equation is y = 2 − 5 x .
Calculating the Value of y Now we want to find the value of y when x = − 6 . Substitute x = − 6 into the equation y = 2 − 5 x :
y = 2 − 5 ( − 6 ) y = 2 − 5 × − 6 y = 2 30 y = 15 Thus, when x = − 6 , y = 15 .
Final Answer The value of y when x = − 6 is 15. Comparing this to the given options, we see that the correct answer is C.
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, your speed is 50 miles per hour. Using direct variation, you can predict that in 3 hours, you would travel 150 miles, assuming the same speed. This principle applies to various situations, such as calculating earnings based on hourly wages or determining the amount of ingredients needed when scaling a recipe.
