To find the equation of the line perpendicular to [tex]\( y = -\frac{1}{2}x - 5 \)[/tex] that passes through the point [tex]\( (2, 7) \)[/tex], let's follow these steps:
1. Identify the slope of the given line:
The equation of the line is [tex]\( y = -\frac{1}{2}x - 5 \)[/tex]. The slope (m) of this 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 slope. The negative reciprocal of [tex]\( -\frac{1}{2} \)[/tex] is [tex]\( 2 \)[/tex]. Thus, the slope of the line we are looking for is [tex]\( 2 \)[/tex].
3. Use the slope-intercept form [tex]\( y = mx + b \)[/tex] to find the y-intercept (b):
We have the slope [tex]\( 2 \)[/tex] and the point [tex]\( (2, 7) \)[/tex] through which the line passes.
Substitute the slope and the coordinates of the point into the slope-intercept form:
[tex]\[
y = 2x + b
\][/tex]
Plug in the point [tex]\( (2, 7) \)[/tex]:
[tex]\[
7 = 2(2) + b
\][/tex]
Simplify this equation to solve for [tex]\( b \)[/tex]:
[tex]\[
7 = 4 + b \\
b = 7 - 4 \\
b = 3
\][/tex]
4. Write the final equation:
Now that we have the slope [tex]\( 2 \)[/tex] and the y-intercept [tex]\( 3 \)[/tex], the equation of the line in slope-intercept form is:
[tex]\[
y = 2x + 3
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\( y = -\frac{1}{2}x - 5 \)[/tex] that passes through the point [tex]\( (2, 7) \)[/tex] is:
[tex]\[ \boxed{y = 2x + 3} \][/tex]
