[br][size=150]The general equation of a straight line is of the form [math]ax+by+c=0[/math] where [math]\left(x,y\right)[/math] denotes a point in the Cartesian plane which sits on the line and the coefficients a,b,c are constants which determine the orientation of the line. [/size]
What makes the equation [math]ax+by+c=0[/math] have a straight graph? How can we be sure?[br][br]Proof:[br]Let [math]A\left(x_1,y_1\right)[/math] , [math]B\left(x_2,y_2\right)[/math] and [math]C\left(x_3,y_3\right)[/math] be any three points on the line. Then we must have [math]ax_1+by_1+c=0[/math], [math]ax_2+by_2+c=0[/math] and[math]ax_3+by_3+c=0[/math]. On subtraction we get [math]a\left(x_2-x_1\right)+b\left(y_2-y_1\right)=0[/math] and [math]a\left(x_3-x_2\right)+b\left(y_3-y_2\right)=0[/math]. Using these two it follows that [math]\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_2}[/math]. This shows that slopes of AB and BC are equal. Hence the proof.
Let us now answer some simple questions.
Which of the coefficients affect the slope of the line?
Why does the line vanish when you put a=0 and b=0 in the input boxes?
The equation fails to carry sense. Because a constant cannot be equal to zero.
Can you now rewrite the general form?
ax+by+c=0 represents a line where a , b , c are real numbers with [math]a^2+b^2\ne0[/math]
Which of the coefficients generates parallel lines?
A is the point where the line meets the x-axis and B is the point where it intersects the y-axis. [br]Which coefficient changes only A?
Set a=0 , b=1 and c=1 in the input boxes . You will get a horizontal line. What is its equation?
Set a=1 , b=0 and c=1 in the input boxes. You will get a vertical line.What is its equation?
For what values of a,b,c you will get a horizontal line at distance of 2 units above x-axis?
For what values of a,b,c you will get a vertical line to the left of y-axis at a distance of 2 units?
Adjust the sliders a and b so that the slope of the line shows m=1. Then adjust c so that the line passes through origin. What is the equation you get?
A straight line is two dimensional. Meaning, we should need only two parameters to describe it . How come we have three in this case namely the coefficients a , b and c?
Can we rewrite the equation so that it will have two coefficients in stead of three?
Yes. We know at least one out of a and b is not zero. Suppose [math]a\ne0[/math]. Then [math]ax+by+c=0\Rightarrow x+\left(\frac{b}{a}\right)y+\left(\frac{c}{a}\right)=0[/math]. Now we can take [math]\frac{b}{a}=\alpha[/math] and [math]\frac{c}{a}=\beta[/math]. This makes it [math]x+\alpha y+\beta=0[/math]. Similarly if [math]b\ne0[/math] we can divide by [math]b[/math] to get [math]\alpha x+y+\beta=0[/math]