Question

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Explanation:

Step1: Recall Direct Variation Definition

Direct variation is a relationship of the form \( y = kx \) (or \( x = ky \)), where \( k \) is a constant, and the graph must pass through the origin \((0,0)\).

Step2: Analyze the Given Graph

The graph shown is a line, but we check if it passes through the origin. From the graph, when \( x = 0 \), \( y = 1 \) (not \( 0 \)). So the error is assuming any line shows direct variation; direct variation lines must pass through the origin.

Step3: Correct the Analysis

To show direct variation, the equation must be \( y = kx \) (passes through \((0,0)\)). Since this line has a \( y \)-intercept of \( 1 \) (not \( 0 \)), it does not represent direct variation. The correct check is: a line represents direct variation only if it passes through the origin (i.e., \( y = kx \) with \( k \) constant and \((0,0)\) on the line).

Answer:

The error is assuming any linear graph shows direct variation. Direct variation graphs (lines) must pass through the origin \((0,0)\). This graph has a \( y \)-intercept of \( 1 \) (does not pass through \((0,0)\)), so \( x \) and \( y \) do not show direct variation. To correct, check if the line passes through \((0,0)\); since it does not, it is not direct variation.