Sure, let's go through this step by step to find the equation of the image of the given line after dilation.
### Step 1: Rewrite the original line equation in slope-intercept form (y = mx + b)
We start with the original line equation:
[tex]\[ 6x + 3y = 3 \][/tex]
First, we solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ 3y = -6x + 3 \][/tex]
[tex]\[ y = -2x + 1 \][/tex]
So the equation of the line in slope-intercept form is:
[tex]\[ y = -2x + 1 \][/tex]
### Step 2: Apply the dilation centered at (0,0)
Since the dilation is centered at [tex]\((0, 0)\)[/tex] with a scale factor of 2, it changes the slope [tex]\( m \)[/tex] to [tex]\( 2m \)[/tex]. The y-intercept remains unchanged because the dilation is centered at the origin.
Given the original slope [tex]\( m = -2 \)[/tex], the new slope [tex]\( m' \)[/tex] after dilation is:
[tex]\[ m' = 2 \times -2 = -4 \][/tex]
### Step 3: Substitute the new slope back into the line equation
We substitute the new slope [tex]\( m' = -4 \)[/tex] and the original y-intercept [tex]\( b = 1 \)[/tex] into the slope-intercept form of the line equation:
[tex]\[ y = -4x + 1 \][/tex]
### Final Equation
The equation of the image of the line after dilation is:
[tex]\[ y = -4x + 1 \][/tex]
Thus, the correct answer is:
c. [tex]\( y = -4x + 1 \)[/tex]
