To determine whether a given graph is the correct representation of the equation \( y = 15x \), we can analyze the properties of this linear equation.
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Slope: The slope \( m \) of the line described by the equation \( y = 15x \) is 15. This means that for every 1 unit increase in \( x \), \( y \) increases by 15 units.
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Y-Intercept: The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation gives \( y = 15 \cdot 0 = 0 \). Thus, the y-intercept is (0, 0), which confirms that this point lies on the line.
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Ordered Pairs: Any point that satisfies the equation \( y = 15x \) should lie on the graph. For example, the point (1, 15) should be on the graph because \( y = 15 \cdot 1 = 15 \).
Now, let’s evaluate the responses based on the above properties:
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The response "Yes, it is the correct graph because the ordered pair (0, 0) satisfies the equation and the slope m = \(\frac{rise}{run} = 15\). " is correct.
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The response "Yes, it is the correct graph because the ordered pair (0, 0) satisfies the equation and the slope m = \(\frac{1}{5}\). " is incorrect; the slope is 15, not \( \frac{1}{5} \).
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The response "Yes, it is the correct graph because the slope = 51." is incorrect; the slope is not 51.
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The response "Yes, it is the correct graph because the slope equals \(\frac{5}{1}\)." is incorrect; the slope is 15.
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The response "No, it is not the correct graph because the slope m = \(\frac{rise}{run} = 15\)." is contradictory, assuming the graph is incorrect because it correctly identifies the slope.
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The response "No, it is not the correct graph because the slope m = \(\frac{1}{5}\)." is incorrect; the slope is not \(\frac{1}{5}\).
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Finally, the response "No, it is not the correct graph because the ordered pair (1, 5) is not a point on the graph of y = 15x." is accurate since the point (1, 5) does not lie on the line; it should be (1, 15).
In conclusion, the correct response is: Yes, it is the correct graph because the ordered pair (0, 0) satisfies the equation and the slope m = \(15\).