What is the rate of change of y with respect to x?
Understand the Problem
The question is asking for the slope of the line represented in the graph, which is the rate of change of y with respect to x. To find the slope, we would need to determine the change in y divided by the change in x.
Answer
The slope is $m = \frac{2}{3}$.
Answer for screen readers
The slope of the line is $m = \frac{2}{3}$.
Steps to Solve
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Identify Points on the Line Locate two clear points on the line. From the graph, we can identify the points (0, 1) and (3, 3).
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Calculate the Change in y Determine the difference in the y-coordinates of the two points: $$ \Delta y = y_2 - y_1 = 3 - 1 = 2 $$
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Calculate the Change in x Determine the difference in the x-coordinates of the two points: $$ \Delta x = x_2 - x_1 = 3 - 0 = 3 $$
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Calculate the Slope The slope $m$ is given by the formula: $$ m = \frac{\Delta y}{\Delta x} $$ So substituting the values we found: $$ m = \frac{2}{3} $$
The slope of the line is $m = \frac{2}{3}$.
More Information
The slope of a line represents the rate of change of $y$ with respect to $x$. In this case, for every increase of 3 units in $x$, $y$ increases by 2 units.
Tips
- Confusing the x-values and y-values when calculating the change, which can lead to an incorrect slope.
- Misreading the graph, leading to incorrect identification of points.
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