Answer :

Answer:

[tex]y=\dfrac{1}{2}x-4[/tex]

Step-by-step explanation:

Perpendicular Lines

Lines that form a 90-degree angle between each other have slopes that are negative reciprocals to each other.

For example,

Line A has a slope of [tex]\dfrac{1}{4}[/tex], so line B's slope is -4.

A more algebraic way of finding the negative reciprocal is,

                                                [tex]-\dfrac{1}{m}[/tex],

where m is line A's slope --or any line's slope.

Finding the Equation of a Line

When given a point on the line, it can be plugged into the linear equation and rearranged to find its m or b value, depending on whether either of the values is known already.

[tex]\hrulefill[/tex]

Solving the Problem

If our equation is perpendicular to a line with a slope of -2, then our equation's m value must be [tex]-\dfrac{1}{-2 } =\dfrac{1}{2}[/tex].

So, our equation is currently,

                                         [tex]y=\dfrac{1}{2}x+b[/tex].

To find the b value we can plug in the given coordinate point,

                                     [tex]-7=\dfrac{1}{2}(-6)+b[/tex]

                                        [tex]-7=-3+b[/tex]

                                             [tex]-4=b[/tex].

So our final answer is,

                                          [tex]y=\dfrac{1}{2}x-4[/tex].

y = (1/2)x - 4 is the equation of the perpendicular line.

The equation of the perpendicular line:

  • Slope of the given line: The given line is in slope-intercept form (y = -2x + 4). The slope of this line is -2.
  • Slope of the perpendicular line: The slope of a line perpendicular to another line is the negative reciprocal of the original slope.  Therefore, the slope of the perpendicular line is 1/2.
  • Point-slope form: We know a point the perpendicular line must pass through (-6, -7) and its slope is 1/2. We can use the point-slope form of linear equations: y - y₁ = m(x - x₁)

where:

y is the y-coordinate of any point on the line

y₁ is the y-coordinate of the known point (-7)

m is the slope (1/2)

x is the x-coordinate of any point on the line

x₁ is the x-coordinate of the known point (-6)

Substitute and solve for y:

y - (-7) = (1/2) (x - (-6))

y + 7 = (1/2)x + 3

Slope-intercept form:

Isolate y to get the equation in slope-intercept form (y = mx + b):

y = (1/2)x + 3 - 7

y = (1/2)x - 4.

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