To determine the equation of the dilated line, let's follow a step-by-step approach:
1. Original Equation in Standard Form:
The given equation of the line is:
[tex]\[
6x + 3y = 3
\][/tex]
2. Convert to Slope-Intercept Form:
To convert the equation into slope-intercept form [tex]\(y = mx + b\)[/tex], we solve for [tex]\(y\)[/tex]:
[tex]\[
3y = -6x + 3
\][/tex]
[tex]\[
y = -2x + 1
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex] and the intercept [tex]\(b\)[/tex] is [tex]\(1\)[/tex].
3. Dilation About the Origin:
When a line is dilated about the origin by a scale factor of [tex]\(2\)[/tex], the slope of the line remains unchanged, but the [tex]\(y\)[/tex]-intercept is scaled by the scale factor.
Slope:
The slope [tex]\(m = -2\)[/tex] remains the same.
Intercept:
The original [tex]\(y\)[/tex]-intercept is [tex]\(b = 1\)[/tex]. After dilation with a scale factor of [tex]\(2\)[/tex], the new intercept becomes:
[tex]\[
b_{\text{new}} = b \times \text{scale factor} = 1 \times 2 = 2
\][/tex]
4. New Equation of the Line:
With the same slope and the new intercept, the equation of the dilated line is:
[tex]\[
y = -2x + 2
\][/tex]
Thus, the correct equation of the image of the line after dilation is:
[tex]\[
\boxed{y = -2x + 2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2. \ y = -2x + 2}
\][/tex]
