To clear the stars, we need to adjust the equation of the line \(y = 4x + 1\) so that it passes through the points where the stars are located. Depending on the stars' coordinates, we can find new slope (m) and/or y-intercept (b) values to achieve this.
Let's first evaluate whether we need to change the slope or the y-intercept. We can check the points provided:
-
For the star at (3,3): \(y = 4(3) + 1 = 12 + 1 = 13\) (not passing through the point)
-
For the star at (6,5): \(y = 4(6) + 1 = 24 + 1 = 25\) (not passing through the point)
-
For the star at (9,7): \(y = 4(9) + 1 = 36 + 1 = 37\) (not passing through the point)
-
For the star at (12,9): \(y = 4(12) + 1 = 48 + 1 = 49\) (not passing through the point)
Since the line does not pass through any of the star coordinates, we need to determine an appropriate new line that passes through these points.
A direct way to find a line that fits all points is to calculate the slope \(m\) using any two star points.
Let's use two points, (3,3) and (12,9).
-
Calculate the slope \(m\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{12 - 3} = \frac{6}{9} = \frac{2}{3} \]
-
Using the point-slope form with point (3,3): \[ y - 3 = \frac{2}{3}(x - 3) \] Expanding this gives: \[ y - 3 = \frac{2}{3}x - 2 \] Then: \[ y = \frac{2}{3}x + 1 \]
The new equation of the line that passes through the stars would be: \[ y = \frac{2}{3}x + 1 \]
With this new equation, the line would now pass through all the star points efficient for clearing them.
If we need further adjustments or just specific results from the original function, feel free to ask!
1 answer