Respuesta :
Answer:
y+4=-(2/5)(x-5) ----> equation of the line into point slope form
y=-(2/5)x-2 ----> equation of the line into slope intercept form
2x+5y=-10 ----> equation of the line in standard form
Step-by-step explanation:
step 1
Find the slope of the given line
we have
5x-2y=-6
isolate the variable y
2y=5x+6
y=2.5x+3
The slope m of the given line is m=2.5
step 2
Find the slope of the line perpendicular to the given line
We know that
If two lines are perpendicular, then their slopes are inverse reciprocal each other
so
m1=5/2
the inverse reciprocal is
m2=-2/5
step 3
Find the equation of the line into point slope form
y-y1=m(x-x1)
we have
m=-2/5
point (5,-4)
substitute
y+4=-(2/5)(x-5) ----> equation of the line into point slope form
y=-(2/5)x+2-4
y=-(2/5)x-2 ----> equation of the line into slope intercept form
Multiply by 5 both sides
5y=-2x-10
2x+5y=-10 ----> equation of the line in standard form
The equations represent the line that is perpendicular to the line 5x − 2y = −6 and passes through the point (5, −4) are as follows:
[tex]y+4=-\frac{2}{5}(x-5)[/tex]
[tex]y=-\frac{2}{5}x-2[/tex]
2x + 5y = - 10
Perpendicular line
Perpendicular lines follows the mathematical principle below.
- m₁m₂ = -1
where
m₁ and m₂ are slopes.
Therefore, using slope intercept form,
y = mx + b
where
m = slope
b = y-intercept
Therefore,
5x - 2y = -6
-2y = -5x - 6
y = 5 / 2 x + 3
The slope is 5 / 2 x
Applying perpendicular rule,
5 / 2m₂ = - 1
m₂ = - 2 / 5
Passing through (5, -4)
-4 = -2/5 (5) + b
-4 + 2 = b
b = - 2
Therefore, the equation is [tex]y=-\frac{2}{5}x-2[/tex]. It can be simplified as follows
5y = -2x - 10
Therefore,
2x + 5y = - 10
using y - y₁ = m (x - x₁) it will be as follows:
[tex]y+4=-\frac{2}{5}(x-5)[/tex]
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