Determine the equation of the line that is perpendicular to the line 5x-4y+3=0 and it passes through the point (-1,2) .

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We'll write the equation in the standard form:

y=mx+n, where m is the slope of the line.

We'll use the property of 2 perpendicular lines, that is:the product between the slopes of 2 perpendicular lines is :-1

We'll note the 2 slopes as m1 and m2.

m1*m2=-1

We could find m1 from the given equation of the line which is perpendicular to the one with the unknown equation.

The equation is 5x-4y+3=0

We'll put the equation into the standard form:

4y=5x+3

We'll divide by 4 both sides:

y=(5/4)x +3/4 => m1=5/4

(5/4)*m2=-1

m2=-4/5

The equation of a line which passes throuh a given point A(-1,2), and it has the slope m2 is:

(y-yA)=m(x-xA)

(y-2)=(-4/5)*(x+1)

We'll remove the brackets and we'll get:

4x + 5y -10+4 = 0

**4x + 5y - 6 = 0**

The equation for the line is:

y-y1 = m (x-x1) where (x1,y1) is any point passes through the line and m is the slope.

We have the point (-1,2) passes through the line.

To calculate the slope (m), we know that the line if perpendicular to the line 5x-4y + 3 = 0. Therefore the product of their slopes should equal -1.

Let us determine the slope for 5x - 4y + 3

First we will rewrite using the slope form>

5x - 4y + 3 = 0

==> 4Y = 5X +3

==> y = (5/4)x + 3/4

Then the slop = 5/4

Then m = -4/5

Then the equation is:

y- y1 = m (x-x1)

y-2 = (-4/5) (x + 1)

y= (-4/5)x - 4/5 + 2

y= (-4/5(x + 6/5

(4/5)x + y - 6/5 = 0

==> **4x + 5y - 6 = 0**

The equation of the given line is:

5x - 4y + 3 = 0

We can convert this equation in the form y = mx + c as follows:

--> 4y = 5x + 3

--> y = (5/4)x = 3/4

In this above equation the slope of the line is equal to value of m. Therefore:

Slope of line = m = 5/4

Slope of a line perpendicular to the first line is equal to:

Slope of perpendicular line = -1/m = -1/(5/4) = -4/5

Therefore equation of the perpendicular line can be written as:

y = -4/5x + c

To find the value of c in the above equation we substitute the coordinates of the given points, that is (-1, 2) in the above equation of perpendicular line:

2 = (-4/5)(-1) + c

--> 2 = 4/5 + c

--> c = 2 - 4/5 = 6/5

Substituting this value of c in equation of perpendicular:

y = (-4/5)x + 6/5

To simplify the equation we multiply both sides by 5, and shift all terms to the left hand side of equation we get:

--> 4x + 5y - 6 = 0

Answer:

Equation of perpendicular line is:

4x + 5y - 6 = 0

Equation of

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