# Determine the equation of a line passing through the points A(-2,6) and B(2,-4). How do i go about this please explain it in steps.

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A(-2,6) B (2,-4)

The equation for the line passing through A and B is:

y- y1 = m (x-x1)

m is the slope:

==> m = (yB-yA)/(xB-xA) = (-4-6)/(-2-2) = -10/-4 = 5/2

Then the equation is:

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

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

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

y= (5/2)x + 1

The equation of a straight line can be written as y=mx+b where m is the slope and b is the y-intercept.

Let the equation of the line between the points A(-2,6) and B(2,-4) be y=mx+b...(1)

We have two pairs of (x,y) values which can be substituted into (1).

For point A(-2,6) we have

6=-2m+b...(2)

and for point B(2,-4) we have

-4=2m+b...(3)

Subtracting (3) from (2) we get 6-(-4)=(-2-2)m+b-b

=>10=-4m

=>m=10/(-4)=-5/2

Substituting m=-5/2 in (2) we get

6=(-5/2)*(-2)+b

=>6=5+b

=>b=1

Now we have the values of m and b with us. We can use these to get the equation of the line as y=(-5/2)x+1

This can also be represented as 2y=-5x+2 or 5x+2y-2=0.

The equation of a line which passes through 2 given points is written as it follows:

(xB - xA)/(x-xA) = (yB - yA)/(y - yA)

xA = -2 and yA = 6

xB = 2 and yB = -4

We'll substitute the coordinates of the given points into the formula written above:

[(2 - (-2)]/[(x-(-2)] = (-4 - 6)/(y - 6)

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

(2+2) / (x+2) = (-10)/(y-6)

4/(x+2)= -10/(y-6)

Now, we'll cross multiply:

-10(x+2) = 4(y-6)

We'll remove the brackets:

-10x - 20 = 4y - 24

We'll move all terms to one side:

-10x - 4y - 20 + 24 = 0

We'll combine like terms and re-arrange the terms:

-4y - 10x + 4 = 0

We'll divide by -2:

2y + 5x - 2 = 0

We've obtained the general form of the equation, that passes through the given points:

**2y + 5x - 2 = 0**

We also could put the equation into the standard form:

2y = -5x + 2

We'll divide by 2 and obtain the standard form:

**y = -5x/2 + 1**

Equation of a line may be expressed as:

y = mx + c

Where:

m = slope of the line

c = a constant

Slope of a line passing through two points (x1, y1) and (x2, y2) is given by the equation:

Slope = (y2 - y1)/(x2 - x1)

To find value of m in the equation of line, we substitute the given values of x1, y1, x2 and y2 in equation for slope:

Slope = m = (-4 - 6)/(2 + 2) = -10/4 = -5/2

To find value of c, we substitute above calculated value of m, and given value of x1 and y1 in general equation of line.

6 = -2(-5/2) + c

==> 6 = 5 + c

c = 6 - 5 = 1

Substituting the calculated values of m and c in the general equation of line, the equation of line passing through the two given point becomes:

y = -5x/2 + 1

This equation can be simplified as:

==> 2y = -5x +1

==> 5x + 2y = 1

The equation of the line passing through the two points (x1,y1) and (x2,y2) is given by:

y-y1 = {(y2-y1)/(x2-x1)}(x-x1).

The given points , (x1,y1) = A(2,6) and (x2,y2) = B(2,-4).

Therefore substitituting these coordinates in the formula (1) we get the equation of the line.

y-6 = {(-4-6)/(2-2)}(x-2}

y-6 = (-10)/0) (x-2). The slope of the line is -10/0 or infinite.

Therefore the given line is perpendicular to x axis or parallel to y axis at x = 2.

So x =2 is the equation of the line.