17JUL21

Ax+By+C=0

Does this represent a straight line?

Yes, it does because it is a "LINEAR" equation in x and y. (The power of x is 1 and the power of y is 1.)

What is the slope of this line?

What is the y-intercept of this line?

And what is the x-intercept?

By = -Ax-C

y = (-Ax-C)/B

y = (-Ax/B) - (C/B)

y = (-A/B)x - (C/B)

Compare this to

y = mx+c

m = -A/B

c = C/B

Therefore, Ax+By+C=0 has a slope of -A/B. Its y-intercept is -C/B. It cuts the Y-Axis at the point (0, -C/B).

To find the x-intercept, we have to substitute 0 for y.

Ax+By+C=0

Ax+B(0)+C=0

Ax+C=0

Ax=-C

x = -C/A

So the x-intercept is -C/A. That is, Ax+By+C=0 cuts the X-Axis at the point (-C/A,0).

Summary:

Ax+By+C=0, Slope=-A/B, y-intercept=-C/B, x-intercept=-C/A

What is the slope of any line parallel to Ax+By+C=0?

-A/B

What is the slope of any line perpendicular to Ax+By+C=0?

Let us say the slope of the perpendicular line is p.

(-A/B)p=-1

p=(-1)(-B/A)

p=B/A


Analyse 2x+3y+4=0.

[Comparing with Ax+By+C=0, A=2, B=3 and C=4.]

01) It's a linear equation in x and y.

02) It represents a straight line.

03) It doesn't pass through the origin.

4A) Its slope: -2/3

4B) It's a ladder leaning against the left wall.

4C) It makes an obtuse angle with the X-Axis.

05) Its y-intercept: -4/3

06) Its x-intercept: -4/2 = -2

07) Slope of a line parallel to it = -2/3

08) Slope of a line perpendicular to it = 3/2

09) (-2,0) is a point on this line.

10) (0,-4/3) is a point on this line.


Let's try to find two more points on this line.

2x+3y+4=0

If x=1,

2(1)+3y+4=0

2+3y+4=0

3y+6=0

3y=-6

y=-6/3=-2

(1,-2) is a point on this line.


If y=1,

2x+3(1)+4=0

2x+3+4=0

2x+7=0

2x=-7

x=-7/2

(-7/2,1) is a point on this line.


Does (4,-4) lie on this line?

2x+3y+4=0

2(4)+3(-4)+4=0

8-12+4=0

0=0

LHS=RHS

Therefore (4,-4) lies on the line.


Does (2,3) lie on this line?

2x+3y+4=0

2(2)+3(3)+4=0

4+9+4=0

17=0

LHS doesn't equal RHS.

So (2,3) doesn't lie on this line.

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