General Equation of a Line: Geometry describes a line as a one-dimensional object that extends infinitely in both directions and has no width. Lines play an important role in the construction of polygons. They are also applied as equations and as graphs in real-life scenarios to make predictions such as weather forecasts, budget prediction and preparation, and calculation of wages. Lines and equations can also be extrapolated to define all possible scenarios around us.

In this article, let’s explore and learn about the general equation of lines, the formula of a straight line, and other amusing factors.

What is a Straight Line?

Geometrically, lines are defined as widthless objects of infinite length. As lines have only length, they are one-dimensional and extend in both directions. Straight lines are defined as the shortest distance between two points, say \(P\) and \(Q\).

We can also say that a line is a figure with zero curvature.

Collinear Points

Two or more points that lie on the same line are said to be collinear.

Observe that points \(P,\,Q\) and \(R\) are collinear points, while \(X,\,Y\) and \(Z\) are non-collinear points.

How do you write equations for a straight line?

There are different ways to write the equation of a straight line. It depends on the facts that we know about the lines provided. The different ways are listed below.

1. Slope-intercept form
2. Intercept form
3. Normal form

Let us now familiarise the general Equation Of A Line before getting into the different writing methods.

What is the equation of a line?

Consider the following coordinates of points whose \(x\) and \(y\) are given below.

\(x\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)
\(y\)	\(9\)	\(10\)	\(11\)	\(12\)	\(13\)

Let’s now plot these points on a graph.

Observe that all these points are collinear; that is, they all fall on a straight line. The graph can be depicted as shown below.

Do you see a pattern in the coordinates of the points?

Observe that the \(y-\)coordinates are \(4\) more than their corresponding \(x-\)coordinates. This fixed relationship of the coordinates of the points that lie on a line forms the equation of the line. This dependency can be represented algebraically as:

\(y = x + 4\) …..(1)

This equation can represent the line, as it holds for every other point on the line.

y-intercept of a Line

The point where the line intersects the \(y-\)axis is called the \(y-\)intercept of the line.

Note that the \(y-\)intercept of the line \(y = x + 4\) is at \((0,\,4)\). At this point, the \(x-\)coordinate is zero.

Algebraically, the \(y-\)intercept of a line is calculated by substituting \(x = 0\) in (1).

\(y = 0 + 4\)

\(y = 4\)

Here, \(y = 4\) is the \(y-\)intercept of the line \(y = x + 4\). It is denoted by \(c\).

Slope of a Line

The steepness of a line with respect to the axes is called the slope of a line. For a straight line, the slope remains constant throughout the line. It also provides information about the direction of the line on the coordinate plane. The slope of a line is determined using the formula:

Slope \( = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\)

where \(\left( {{x_1},\,{y_1}} \right),\;\left( {{x_2},\,{y_2}} \right)\) coordinates of two points on the line

Observe that the slope is the ratio of similar units. Hence, it is dimensionless; that is, the slope of a line has no units. It is denoted by \(m\).

For the given line, consider any two points, say: \((1,\,5)\) and \((8,\,12)\)

\(\therefore \,m = \frac{{12 – 5}}{{8 – 1}}\)

\(m = \frac{7}{7} = 1\)

Slope of a line is also called its gradient.

Slope-intercept Form of a Line

Now that we have familiarised the \(y-\)intercept and slope of a line, let us see how it varies with different lines.
Consider the lines \(y = x,\,y = 2x,\,y = 2x + 1\), and \(y = 2x – 1\). On a graph, they will look as shown below.

Let us now tabulate the \(y-\)intercept and slope of these lines.

Equation	\(y\)-intercept	Slope
\(y = x\)	\(c = 0\)	\(m = 1\)
\(y = 2x\)	\(c = 0\)	\(m = 2\)
\(y = 2x + 1\)	\(c = 1\)	\(m = 2\)
\(y = 2x – 1\)	\(c = – 1\)	\(m = 2\)

Do you notice the similarity in the equations of the lines?

The equation of a straight line with slope \(m\) and \(y-\)intercept \(c\) is given by \(y = mx + c\).

So, in any linear equation of the form \(y = mx + c\), the coefficient of \(x\) is the slope, and the constant in the equation is its intercept on the \(y-\)axis.

We can also work backwards on this one. Let’s say the slope, \(m = \frac {1}{2}\) and the \(y-\) intercept, \(c = – \frac {2}{3}\). Then, to find the equation, we must simply substitute these values in the general equation of a straight line, \(y = mx + c\).

Therefore, the equation of the line with a slope, \(m = \frac {1}{2}\) and the \(y-\) intercept, \(c = \frac {2}{3}\) is \(y = \frac{1}{2}x – \frac{2}{3}\).

This equation can also be written as \(6y = 3x – 4\) or \(3x – 6y = 4\).

Need for General Equation of a Line

In the slope-intercept equation of a line \(y = mx + c\),

\(m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\) \(\to\) slope

\(c \to y-\) intercept

For a vertical line, we know that the \(x-\)coordinate is \(0\). Hence, the calculation of the slope of a vertical line will have a zero as the denominator. This means that the slope of a vertical line is undefined and hence cannot be written in the form of \(y = mx + c\).

Thus, arises the need for a more general equation for straight lines.

General Equation of a Line

The graph of a linear equation is always a straight line. Hence, the most commonly used equation of straight lines is also a linear polynomial of the form:

\(Ax + By + C = 0\)

This is called the general equation of a straight line.

Case 1:

When \(x = 0\), we get \(By + C = 0\).

\( \Rightarrow y = – \frac{C}{B}\)

For all values of \(C\) and \(B\), the lines are horizontal and parallel to the \(x-\)axis.

Case 2:

When \(y = 0\), we get \(Ax + C = 0\).

\( \Rightarrow x = – \frac{C}{A}\)

For all values of \(C\) and \(B\), the lines are vertical and parallel to the \(y-\)axis.

Let’s learn the various forms of the general equation of a line.

1. Slope-intercept form

In the general equation, \(Ax + By + C = 0\),

Condition	when \(B \ne 0\),	when \(B = 0\),
Simplification	\(Ax + By = – C\) \(y = – \frac{A}{B}x – \frac{C}{B}\)	\(Ax + C = 0\) \(x = – \frac{C}{A}\)
Nature of line	Straight line	Vertical line
Slope	\(m = – \frac{A}{B}\)	Undefined
Intercept	\(y = – \frac{C}{B}\)	\(x = – \frac{C}{A}\)

2. Intercept form

In the general equation, \(Ax + By + C = 0\),

Condition	when \(C \ne 0\),	when \(C = 0\)
Simplification	\(\frac{x}{{ – \frac{C}{A}}} + \frac{y}{{ – \frac{C}{B}}} = 1\) Let, \( – \frac{C}{A} = a\) and \( – \frac{C}{B} = b\) \(\therefore \,\frac{x}{a} + \frac{y}{b} = 1\)	\(Ax + By = 0\) \(y = – \frac {A}{B} x\)
Nature of line	Straight line	Straight line
Intercept	\(y = b = – \frac{C}{B}\) \(x = a = – \frac{C}{A}\)	No intercepts, as the line passes through the Origin

3. Normal form

Let \(x\cos \;\omega + y\sin \;\omega = p\) be the normal form of the line, which is the same as \(Ax + By + C = 0\).
\(\therefore \,\frac{A}{{\cos \;\omega }} = \frac{B}{{\sin \;\omega \;}} = \frac{{ – C}}{p}\)
Equating, we get,

\(\cos \,\omega = \, – \frac{{Ap}}{C}\)

\(\sin \,\omega = \, – \frac{{Bp}}{C}\)

Using the trigonometric identity, \(\theta + \theta = 1\), we get,

\({\left( { – \frac{{Ap}}{C}} \right)^2} + {\left( { – \frac{{Bp}}{C}} \right)^2} = 1\)

\( \Rightarrow {p^2} = \frac{{{C^2}}}{{{A^2} + {B^2}}}\)

\(\therefore \,p = \pm \frac{C}{{\sqrt {{A^2} + {B^2}} }}\)

Here, make a proper choice of signs, plus or minus, such that \(p\) is always positive.

Substituting for \(p\), we get,

\(\cos \,\omega = \pm \frac{A}{{\sqrt {{A^2} + {B^2}} }}\)

\(\sin \;\omega = \pm \frac{B}{{\sqrt {{A^2} + {B^2}} }}\)

Steps to Convert General Equation to Normal Form

Step 1: Move the constant term to the right side of the equation. Keep it positive.
The equation is now of the form: \(Ax + By = C\)
Step 2: Divide both sides of the equation by \(\sqrt {{A^2} + {B^2}} \).
Here,
\(A \to\) Coefficient of \(x\)
\(B \to\) Coefficient of \(y\)
The resulting equation is the normal form.

Solved Examples on General Equation of a Line

Q.1. The equation of a line is \(3x + 5y = 15\). Find its slope and intercepts.
Ans:

Slope (m)	Intercepts
Given: \(3x + 5y = 15\) \(\Rightarrow y = \frac{{15 – 3x}}{5}\) \(\therefore \,y = 5 – \frac{3}{5}x\) This is of the form: \(y = mx + c\) Here, slope, \(m = – \frac {3}{5}\)	Given: \(3x + 5y = 15\) \( \Rightarrow \frac{x}{5} + \frac{y}{3} = 1\) This is of the form: \(\frac{x}{a} + \frac{y}{b} = 1\) Here, \(x-\)intercept, \(a = 5\) \(y-\)intercept, \(b = 3\)

Q.2. Find the equation of the line.

Ans: From the graph,
\(x-\)intercept, \(a = 2\)
\(y-\)intercept, \(b= – 1\)
The intercept form of a line is given by: \(\frac{x}{a} + \frac{y}{b} = 1\)
\(\therefore \,\frac{x}{2} + \frac{y}{{ – 1}} = 1\)
\(\frac{x}{2} – y = 1\)
This can be written as: \(y = \frac{x}{2} – 1\).

Q.3. Find the equation of the line that cuts the positive \(x-\)axis at \(3\) and the negative \(y-\)axis at \(8\).
Ans: Given: \(x-\)intercept, \(a = 3\)
\(y-\)intercept, \(b = – 8\)
Therefore the equation of the line is: \(\frac{x}{3} – \frac{y}{8} = 1\)
Simplifying, we get,
\(8x – 3y = 24\)
\( \Rightarrow 8x – 3y – 24 = 0\).

Q.4. Transform \(3x + 4y = 5\surd 2\) to normal form.
Ans: Given: \(3x + 4y = 5\sqrt 2\)
Here,

\(A = 3\)

\(B = 4\)

\(C = – 5\sqrt 2 \)

\(p = \pm \frac{C}{{\sqrt {{A^2} + {B^2}} }}\)
\(\therefore \,p = \pm \frac{{ – 5\sqrt 2 }}{{\sqrt {{3^2} + {4^2}} }} = \pm \frac{{5\sqrt 2 }}{5}\)
\( \Rightarrow p = \sqrt 2 \)

\(\cos \,\omega = \, – \frac{{Ap}}{C}\) \(\cos \,\omega = \, – \frac{{3\sqrt 2 }}{{ – 5\sqrt 2 }}\) \(\cos \,\omega = \,\frac{3}{5}\)

\(\sin \,\omega = \, – \frac{{Bp}}{C}\) \(\sin \,\omega = \, – \frac{{4\sqrt 2 }}{{ – 5\sqrt 2 }}\) \(\sin \,\omega = \,\frac{4}{5}\)

Hence, the normal form of the equation is: \(\frac{3}{5}x + \frac{4}{5}y = \sqrt 2 \).

Q.5. Reduce to slope-intercept form: \(- 5x + 2y = 9\). Find its slope.
Ans: Given: \(- 5x + 2y = 9\)
Rearranging, we get,
\(2y = 9 + 5x\)
Slope-intercept form, \(y = \frac{5}{2}x + \frac{9}{2}\)
This is of the form: \(y = mx + c\)
Hence, the slope, \(m = \frac{5}{2}\).

Summary

A line is a geometric object that has infinite length and no width. There are different ways to write the equation of straight lines depending on the known factors. When slope \((m)\) and \(y-\) intercept \((c)\) of a line are known, the equation is defined as \(y = mx + c\). However, for a vertical line, the slope is undefined.
Hence, we need a different form of equation called the general equation of a straight line. It is given by: \(Ax + By + C = 0\). There are three different ways to write this general equation: the slope-intercept form, the intercept form, and the normal form. Solved examples help us understand how to employ the different forms of the general equation.

FAQs on Equation of a Line

Q.1. How do you find the general equation of a line?
Ans: The general equation of a line when slope and y-intercept are given is \(y = mx + c\). Here, \(m\) is the slope of the line and is defined as \(\frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\) and \(c\) is the intercept on the \(y-\)axis.

Q.2. What is the general form of a line?
Ans: A line can be defined in different forms, namely the intercept form and the slope-intercept form. However, the need for a general form arises when neither of the values is defined. The general form of a line is \(Ax + By + C = 0\). This is also called the general equation of the straight line.

Q.3. Is general and standard form the same for straight lines?
Ans: The general and standard forms of a straight line are two equivalent methods of defining the linear function. While the general form of a line is \(Ax + By + C = 0\), the standard form is given by: \(y = mx + c\), where \(m\) is the slope and \(y\) is the intercept of the line on the \(y-\)axis.

Q.4. What is an example of an equation of a line?
Ans: All polynomials whose degree is one is linear. This means that their graph is a straight line. Examples: \(y – 4x = 1\), \(\frac{2}{x} + \frac{6}{y} = 15\), and \(x = -7.5\).

Q.5. How do you graph a general equation of a straight line?
Ans: Steps to graph a linear equation of the form: \(Ax + By + C = 0\):
Step 1: Find the \(y-\)intercept, using \(x = 0\). Plot on a graph.
Step 2: Find the \(x-\)intercept, using \(y = 0\). Plot the coordinate.
Step 3: Connect the intercepts using a straight line, and show its extension on both sides.

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