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Mathemati cs Title: Equations of Straight Line O y x
Mathematics
Title: Equations of Straight Line
O
y
x
x 22 , yxBy
Ox
x
11, yxA
x yxP ,
Two-Point Form
Given a straight line which passes through the points A and B, then
12
12
xx
yy
Slope of AB =
If P(x, y) is any point on the line AB, then
1
1
xx
yy
Slope of PA =
Since PA and AB are parts of the same straight line, then
Slope of PA = Slope of AB
12
12
xx
yy
1
1
xx
yy
=
This equation is known as the Two-Point Form of the straight line. (Go to Example 1)
i.e.
Point-Slope Form
x
y
Ox
11, yxA
x
yxP ,Slope = m
Given a straight line which passes through the point A and has m as its slope.
If P(x, y) is any point on the line , then
1
1
xx
yy
Slope of PA =
Since slope of PA equals to the slope of the straight line, then
1
1
xx
yy
= m
11 xxmyy i.e.
This equation is known as the Point-Slope Form of the straight line. (Go to Example 2)
O
y
x
c
A(0, c)x
xP(x, y)
Slope = m
Slope-Intercept Form
Given a straight line which cuts the y-axis at A and with slope m .
(Note: c is called the y-intercept of the straight line.)
If P is any point on the line, then
Slope of PA = 0
x
cy
Since slope of PA is equal to the slope of the line, then by Point-Slope Form
(y - c) = m(x – 0)
y = mx + ci.e.
This equation is known as the Slope-Intercept Form of the straight line. (Go to Example 3)
Intercept Form
x
xa
b
Given a straight line which cuts the x-axis at A and y-axis at B.
(Note: a is called the x-intercept of the straight line.)
x
y
Ox
B(0, b)
A(a, 0)
P(x, y)
=
a
b
0
0Slope of AB =
a
b
ax
y
0
Slope of PA =
If P is any point on the line, then
Since the slope of PA equals to the slope of AB, then
ax
y
0
a
b=
bx + ay = abDividing both sides by ab, we have
1b
y
a
x
This equation is known as the Intercept Form of the straight line. (Go to Example 4)
Standard Form
It should be noted that all the different standard forms of the equation of a straight line can be reduced to the form
Ax + By + C = 0
where A, B and C are constants with A and B not both zero.
This equation is known as the Standard Form of a straight line.
Mathematics
Linear Equations
O
y
x
Example 1 (Two-Point Form)
y
Ox
x
x yxP ,
x
A(1, 3)
B(5, 6)Find the equation of the straight line.
The required equation:
15
36
1x
3y
4(y – 3) = 3(x – 1)
i.e. 3x – 4y + 9 = 0
(Note: The answer is in General Form)
Example 2 (Point-Slope Form)
y
Ox
Slope = 2 P(x, y)x
A(1, 3)x
Find the equation of the straight line.
The required equation:
y – 3 = 2(x – 1)
i.e. 2x – y + 1 = 0
(Note: The answer is in General Form)
Example 3 (Slope-Intercept Form)
xA(0, 4)
Slope = 2
Find the equation of the straight line.
The required equation:
y = 2x + 4
i.e. 2x – y + 4 = 0
(Note: The answer is in General Form)
y
xO
xP(x, y)
Example 4 (Standard Form)
y
Ox
B(0, 5)
A(7, 0)
P(x, y)
x
x
x
Find the equation of the straight line.
The required equation:
15
y
7
x
5x + 7y = 35
i.e. 5x + 7y – 35 = 0
(Note: The answer is in Standard Form)
