To modify the equation of the line \( y = \frac{1}{2}x + 1 \), you can adjust either the slope or the y-intercept to change the trajectory of the line.
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Changing the Slope: The slope determines how steep the line is. If the slope is too low (as in this case, \(\frac{1}{2}\)), you might want to increase it to make the line steeper. For example, changing the slope to 1 would give:
\[ y = x + 1 \]
or increasing it further to 2 would yield:
\[ y = 2x + 1 \]
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Changing the Y-Intercept: The y-intercept affects where the line crosses the y-axis. If you need the line to be higher or lower, you can adjust this value. For example, changing the y-intercept from +1 to +3 gives:
\[ y = \frac{1}{2}x + 3 \]
or changing it to -1 would yield:
\[ y = \frac{1}{2}x - 1 \]
To specifically determine what changes you need to make to clear the stars, it's important to know the position of the stars relative to the line. You can experiment with both the slope and the y-intercept to find the best configuration for clearing the stars in the marble slide scenario.