Question 3 of 5

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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]. Write this line in slope-intercept form.

[tex]y = [/tex]

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Answer :

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]