To determine the slope of a line that is perpendicular to the given line, we first need to find the slope of the given line. Here’s the detailed step-by-step solution:
1. Convert the given line equation to slope-intercept form:
The equation provided is [tex]\(6x + 3y - 1 = 0\)[/tex]. To find the slope, we need to express this equation in the slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
2. Isolate [tex]\(y\)[/tex] in the equation:
Start by moving the term involving [tex]\(x\)[/tex] and the constant to the other side of the equation:
[tex]\[
6x + 3y - 1 = 0 \quad \Rightarrow \quad 3y = -6x + 1
\][/tex]
3. Divide all terms by the coefficient of [tex]\(y\)[/tex]:
To solve for [tex]\(y\)[/tex], divide each term by 3:
[tex]\[
y = \frac{-6x}{3} + \frac{1}{3}
\][/tex]
Simplifying this, we get:
[tex]\[
y = -2x + \frac{1}{3}
\][/tex]
4. Identify the slope of the given line:
Now, we can see that the slope-intercept form of the line is [tex]\(y = -2x + \frac{1}{3}\)[/tex]. Therefore, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-2\)[/tex].
5. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line’s slope. The negative reciprocal of [tex]\(-2\)[/tex] is:
[tex]\[
\frac{-1}{-2} = \frac{1}{2}
\][/tex]
Hence, the slope of any line perpendicular to the given line [tex]\(6x + 3y - 1 = 0\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
