Certainly! Let's solve the problem step-by-step:
1. Identify the slope of the given line:
The equation of line [tex]\( CD \)[/tex] is [tex]\( y = 3x - 3 \)[/tex]. Here, the slope ([tex]\( m \)[/tex]) of line [tex]\( CD \)[/tex] is 3.
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Thus, the slope of the line perpendicular to [tex]\( CD \)[/tex] is:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{3}
\][/tex]
3. Use the point-slope form of the equation:
We need to find an equation of the line with the perpendicular slope that passes through the point [tex]\( (3, 1) \)[/tex]. The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\( (3, 1) \)[/tex] and [tex]\( m \)[/tex] is the slope [tex]\( -\frac{1}{3} \)[/tex].
4. Substitute the point and the perpendicular slope into the point-slope form:
[tex]\[
y - 1 = -\frac{1}{3} (x - 3)
\][/tex]
5. Simplify the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 1 = -\frac{1}{3}x + 1
\][/tex]
Adding 1 to both sides:
[tex]\[
y = -\frac{1}{3}x + 2
\][/tex]
So, the equation of the line in slope-intercept form that is perpendicular to [tex]\( CD \)[/tex] and contains the point [tex]\( (3, 1) \)[/tex] is:
[tex]\[
y = -\frac{1}{3}x + 2
\][/tex]
Therefore, among the provided options, the correct equation of the line in slope-intercept form is not listed. The correct equation should be:
[tex]\[
y = -\frac{1}{3}x + 2
\][/tex]
