A line is drawn through $(-4,3)$ and $(4,3)$. Which describes whether or not the line represents a direct variation?
A. The line represents a direct variation because $-\frac{4}{3}=\frac{4}{3}$.
B. The line represents a direct variation because it is horizontal.
C. The line does not represent a direct variation because it does not go through the origin.
D. The line does not represent a direct variation because $-4(3) \neq 4(3)$.
Answer (2)
The line passes through ( − 4 , 3 ) and ( 4 , 3 ) , so it's a horizontal line with the equation y = 3 .
A direct variation must pass through the origin ( 0 , 0 ) .
Since the line y = 3 does not pass through the origin, it is not a direct variation.
Therefore, the line does not represent a direct variation because it does not go through the origin. The line does not represent a direct variation because it does not go through the origin.
Explanation
Understanding Direct Variation First, let's analyze the given information. We have a line that passes through the points ( − 4 , 3 ) and ( 4 , 3 ) . We need to determine if this line represents a direct variation. A direct variation is a linear relationship of the form y = k x , where k is a constant of proportionality. A key characteristic of direct variation is that the line must pass through the origin ( 0 , 0 ) .
Finding the Equation of the Line To determine if the given line represents a direct variation, we first need to find the equation of the line. Since both points have the same y -coordinate, the line is a horizontal line. The equation of the line is y = 3 .
Checking if the Line Passes Through the Origin Now, let's check if the line passes through the origin ( 0 , 0 ) . If we substitute x = 0 into the equation y = 3 , we get y = 3 , not 0 . Therefore, the line does not pass through the origin.
Conclusion Since the line does not pass through the origin, it cannot be represented in the form y = k x . Therefore, the line does not represent a direct variation.
Examples
Direct variation is a fundamental concept in many real-world scenarios. For example, the distance you travel at a constant speed is directly proportional to the time you spend traveling. If you travel at 60 miles per hour, the equation is d = 60 t , where d is the distance and t is the time. This equation represents a direct variation because the distance is directly proportional to the time, and the graph of this equation passes through the origin. Another example is the relationship between the number of items you buy and the total cost, assuming each item has the same price. If each item costs 5 , t h e t o t a l cos t C i s g i v e nb y C = 5n , w h ere n$ is the number of items. This is also a direct variation.
The line through the points (-4, 3) and (4, 3) is horizontal and given by the equation y = 3. Since this line does not pass through the origin (0, 0), it does not represent a direct variation. Thus, the correct answer is that the line does not represent a direct variation because it does not go through the origin.
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