Answer:
1) The image has the same slope, different y-intercept
Explanation:
You want the relationship between the line 3x -5y = 4 and its image dilated by a factor of 5/3 centered at the origin.
Dilation
Dilation about the origin multiplies every coordinate by the dilation factor. Dilation does not change slopes or angles, but it changes lengths by the dilation factor.
Dilation of a line
When the equation of a line is written in standard form, the dilation of it effectively multiplies the constant by the dilation factor.
The given line has intercepts ...
ax +by = c
x-intercept = c/a = 4/3
y-intercept = c/b = -4/5
If these values are multiplied by the dilation factor 5/3, they become ...
x-intercept' = (5/3)(4/3) = 20/9 = 2 2/9
y-intercept' = (5/3)(-4/5) = -4/3 . . . . . note the y-intercept has changed
Slope
The slope of the given line can be determined from its intercepts:
m = (-y intercept)/(x intercept) = -(-4/5)/(4/3) = 3/5
Since both of these intercepts are multiplied by the same dilation factor, their ratio remains unchanged. The slope of the dilated line is the same.
Equation
Effectively, the dilated line's equation is ...
3x -5y = 4(5/3)
3x -5y = 20/3
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Additional comment
The original (dashed) line and its dilated image are shown in the attachment.