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Sagot :
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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