Certainly! Let's go through the process of finding the equation of the dilated line step-by-step:
1. Initial Equation:
The original line is given by the equation [tex]\(6x + 3y = 3\)[/tex].
2. Understanding Dilation:
Dilation of a line centered at the origin (0,0) by a scale factor of 2 means that we're effectively changing the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] (and the constant term) by dividing them by the scale factor because dilation stretches or shrinks distances from the origin.
3. Divide Coefficients by Scale Factor:
The scale factor given is 2, so we need to divide the coefficients [tex]\(6\)[/tex], [tex]\(3\)[/tex], and the constant term [tex]\(3\)[/tex] by 2.
- Coefficient of [tex]\(x\)[/tex]:
[tex]\[
\frac{6}{2} = 3.0
\][/tex]
- Coefficient of [tex]\(y\)[/tex]:
[tex]\[
\frac{3}{2} = 1.5
\][/tex]
- Constant term:
[tex]\[
\frac{3}{2} = 1.5
\][/tex]
4. Formulating the New Equation:
The new equation of the line after dilation will thus have the coefficients we just calculated:
[tex]\[
3.0x + 1.5y = 1.5
\][/tex]
For simplicity, this can be rewritten without the decimal points (if desired), though it's perfectly correct as is. So, the equation of the dilated line is:
[tex]\[
3.0x + 1.5y = 1.5
\][/tex]
So, the detailed step-by-step solution confirms that the equation of the image of the line after dilation by a scale factor of 2 centered at the origin [tex]\((0,0)\)[/tex] is:
[tex]\[
3.0x + 1.5y = 1.5
\][/tex]
