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EQUATION OF THE FIRST DEGREE, THE STRAIGHT LINE

The general form of an equation of the first degree in two variables is

ax + by + c = 0 .

Its locus is a straight line. Every straight line in the plane can be represented by a first degree equation in two variables. Every first degree equation in two variables is represented by a straight line in the plane.

Special forms of the equation of a straight line.

1] Point-slope form. The equation of a straight line that passes through a point P_{1}(x_{1}, y_{1}) with a
slope m is

y - y_{1} = m(x - x_{1})

2] Point-intercept form. The equation of a straight line having slope m and y intercept b is

y = mx + b

3] Two-point form. The equation of the straight line passing through points P_{1}(x_{1}, y_{1}) and P_{2}(x_{2},
y_{2}) is

4] Intercept form. The equation of the straight line whose x and y intercepts are a and b, respectively, is

5] Normal form. The normal form of the equation of a straight line is

x cos α + y sin α - p = 0

where α is the angle from the x-axis to the perpendicular from the origin to the line and p is the length of the perpendicular. See figure 1.

General form. The general form of the equation of a straight line is

ax + by + c = 0

where a, b and c are arbitrary constants. This form includes all other forms as special cases. For an equation in this form the slope is -a/b and the y intercept is -c/b. An equation in general form can be changed to normal form by dividing by

where the sign of the radical is taken opposite to that of c if c 0 and the same as that of b if c = 0. Thus the normal form of ax + by + c = 0 is

Perpendicular distance from a line to a point. The perpendicular distance d from the
line ax + by + c = 0 to the point P_{1}(x_{1}, y_{1}) is given by

where the sign of the radical is taken opposite to that
of c if c
0 and the same as that of b if c = 0. The
distance d is positive if P_{1} is on the opposite side of
the line from the origin and negative if it is on the
same side of the line as the origin.

Parallel lines.

1] The two lines y = m_{1}x + b_{1} and y = m_{2}x + b_{2} are parallel if m_{1} = m_{2}.

2] The two lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 are parallel if a_{1 }/a_{2} = b_{1} /b_{2}.

Equation of a line through a point parallel to a given line. The equation of a line through the
point P_{1}(x_{1}, y_{1}) and parallel to line ax + by + c = 0 is

a(x - x_{1}) + b(y - y_{1}) = 0

Perpendicular lines.

1] The two lines y = m_{1}x + b_{1} and y = m_{2}x + b_{2} are perpendicular if m_{1} = -1/m_{2}.

2] The two lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2 } = 0 are perpendicular if a_{1}a_{2} + b_{1}b_{2} =
0.

Equation of a line through a point perpendicular to a given line. The equation of a line
through the point P_{1}(x_{1}, y_{1}) and perpendicular to line ax + by + c = 0 is

b(x - x_{1}) + a(y - y_{1}) = 0

Intersecting lines.

1] Let a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 be the equations of two intersecting lines and λ
an arbitrary constant. Then

(a_{1}x + b_{1}y + c_{1} ) + λ(a_{2}x + b_{2}y + c_{2 }) = 0

represents the system of lines through the point of intersection.

2] The three lines a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2 } = 0, and a_{3}x + b_{3}y + c_{3} = 0 meet in a
point if

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