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], follow these steps:
1. Identify the slope of the given line: The given line is in slope-intercept form [tex]\( y = mx + b \)[/tex]. Here, [tex]\( m = -\frac{1}{2} \)[/tex].
2. Find the slope of the perpendicular line: For a line to be perpendicular to another, its slope must be the negative reciprocal of the original slope. The negative reciprocal of [tex]\( -\frac{1}{2} \)[/tex] is 2.
Therefore, the slope of the perpendicular line is [tex]\( m = 2 \)[/tex].
3. Use the slope-intercept form equation [tex]\( y = mx + b \)[/tex]: We now have the slope [tex]\( m = 2 \)[/tex] and need to find the y-intercept [tex]\( b \)[/tex]. We will use the point [tex]\( (2, 7) \)[/tex] which the line passes through.
4. Substitute the point into the equation: Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 7 \)[/tex] into the slope-intercept form equation to solve for [tex]\( b \)[/tex]:
[tex]\[
7 = 2(2) + b
\][/tex]
5. Solve for [tex]\( b \)[/tex]:
[tex]\[
7 = 4 + b
\][/tex]
[tex]\[
b = 3
\][/tex]
6. Write the equation of the line: Now that we have the slope [tex]\( m = 2 \)[/tex] and the y-intercept [tex]\( b = 3 \)[/tex], the equation of the line 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]\[ y = 2x + 3 \][/tex]
