Equation of a line
Introduction
An equation of a line is a mathematical statement that describes all the points that lie on that particular line in a coordinate plane. It provides a relationship between the x-coordinate (abscissa) and y-coordinate (ordinate) of every point on the line.
In simpler terms, if we plug any x-value into the equation, it gives us the corresponding y-value for a point on the line, and vice versa. The equation of a line helps us understand the line's slope (its steepness and direction) and intercepts (where it crosses the x-axis and y-axis).
Various forms of the equation of a line
-
Slope-Intercept Form
The slope-intercept form of a line's equation is given by:
$$ y = mx + b $$
where,
- 𝑦 is the dependent variable (ordinate)
- 𝑥 is the independent variable (abscissa)
- 𝑚 is the slope of the line
- 𝑏 is the y-intercept (the point where the line crosses the y-axis)
Example:
Consider the equation
$$ y = 2x + 3 $$
In this equation, the slope
𝑚
is 2, and the y-intercept
𝑏
is 3. This means that for every 1 unit increase in
𝑥
,
𝑦
increases by 2 units, and the line crosses the y-axis at the point (0, 3).
-
Point-slope form
The point-slope form of a line's equation is given by:
$$
𝑦
−
𝑦
1
=
𝑚
(
𝑥
−
𝑥
1
)
$$
where:
(
𝑥
1
,
𝑦
1
)
is a point on the line
𝑚
is the slope of the line
Example:
Consider the point (4, 5) and a slope
𝑚
of 3.
Using the point-slope form, the equation of the line becomes:
$$
𝑦
−
5
=
3
(
𝑥
−
4
)
$$
This equation represents a line that passes through the point (4, 5) with a slope of 3.
-
Two-Point Form
The two-point form of a line's equation is used when two points on the line are known. It is given by:
$$
\frac{
𝑦
−
𝑦_
1
}{
𝑦_
2
−
𝑦_
1
}
=
\frac{
𝑥
−
𝑥_
1
}{
𝑥_
2
−
𝑥_
1
}
$$
where
(
𝑥
1
,
𝑦
1
)
and
(
𝑥
2
,
𝑦
2
)
are two points on the line.
Example:
Consider the points (2, 3) and (5, 7).
Using the two-point form, the equation of the line becomes:
$$
\frac{
𝑦
−
3
}{
7
−
3
}
=
\frac{
𝑥
−
2
}{
5
−
2
}
$$
Simplifying this equation, we get:
$$
\frac{
𝑦
−
3
}{
4
}
=
\frac{
𝑥
−
2
}{
3
}
$$
Cross-multiplying, we get the linear equation:
$$
3(y - 3) = 4(x - 2)
$$
$$
3y - 9 = 4x - 8
$$
$$
4x - 3y = -1
$$
-
Doduble Intercept Form
The intercept form of a line's equation is given by:
$$
\frac
{𝑥}
{𝑎}
+
\frac
{𝑦}
{𝑏}
=
1
$$
where
𝑎
and
𝑏
are the x-intercept and y-intercept, respectively.
Example:
Consider the x-intercept
𝑎
=
4
and y-intercept
𝑏
=
2
.
Using the double intercept form, the equation of the line becomes:
$$
\frac
{𝑥}
{4}
+
\frac
{𝑦}
{2}
=
1
$$
Multiplying through by 4 to clear the denominator, we get:
$$ x + 2y = 4 $$
-
Standard Form
The standard form of a line's equation is given by:
$$
𝐴
𝑥
+
𝐵
𝑦
=
𝐶
$$
where:
𝐴
,
𝐵
, and
𝐶
are integers (with
𝐴
and
𝐵
not both zero)
Example:
Consider the equation
$$
3
𝑥
+
4
𝑦
=
12
.
$$
In this form, it is often useful for solving systems of linear equations and performing algebraic manipulations.