First, let's rewrite the equations of the two lines in slope-intercept form:
y = (1/2)x + 3/2 (for y_2x = 3)
y = (4/2)x + 9/2 = 2x + 9/2 (for 2y_x = 9)
To find the axis of reflection, we need to find the equation of the line that is perpendicular to y_2x = 3 and passes through its midpoint. This line will be the axis of reflection.
The slope of y_2x = 3 is -2, so the slope of the perpendicular line is 1/2. The midpoint of y_2x = 3 is at (0, 3/2). Using the slope-intercept form of a line, the equation of the perpendicular line is:
y - 3/2 = 1/2(x - 0)
Simplifying, we get:
y = (1/2)x + 3/2
This is the equation of the axis of reflection. We can confirm that the line y_2x = 3 is reflected across this line to become 2y_x = 9 by checking that the distance between each point on the original line and the axis of reflection is equal to the distance between its corresponding point on the reflected line and the axis of reflection.
In a reflection the image of the line y_2x=3 is the line 2y_x=9. Find the axis of reflection.
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