To write the equation of a line perpendicular to the given line [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] that passes through the point [tex]\((-5, -1)\)[/tex], follow these steps:
1. Find the Slope of the Given Line:
The slope of the given line is [tex]\(-\frac{1}{2}\)[/tex].
2. Determine the Slope of the Perpendicular Line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
Hence, the slope of the line perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] is:
[tex]\[
m = -\left(\frac{-1}{2}\right)^{-1} = 2
\][/tex]
3. Use the Point-Slope Form of the Equation:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope. Here, [tex]\((x_1, y_1) = (-5, -1)\)[/tex] and [tex]\(m = 2\)[/tex].
Plugging in the values, we have:
[tex]\[
y - (-1) = 2(x - (-5))
\][/tex]
4. Simplify the Point-Slope Form:
Simplify the equation from the previous step:
[tex]\[
y + 1 = 2(x + 5)
\][/tex]
5. Convert to Slope-Intercept Form:
The slope-intercept form of a line's equation is:
[tex]\[
y = mx + b
\][/tex]
We will distribute and solve for [tex]\(y\)[/tex] in the previous form:
[tex]\[
y + 1 = 2x + 10
\][/tex]
Subtract 1 from both sides:
[tex]\[
y = 2x + 9
\][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[
y = 2x + 9
\][/tex]
To summarize, the equation of a line perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passing through the point [tex]\((-5, -1)\)[/tex] is:
Point-Slope Form:
[tex]\[
y - (-1) = 2(x - (-5))
\][/tex]
Slope-Intercept Form:
[tex]\[
y = 2x + 9
\][/tex]
