Click to Chat
1800-2000-838
+91-120-4616500
CART 0
Use Coupon: CART20 and get 20% off on all online Study Material
Welcome User
OR
LOGIN
Complete Your Registration (Step 2 of 2 )
General Equation of a Line
Who is the founder of Linear Equation?
What is the definition of Linear Equation in math?
How do you describe a Linear Equation?
What is a Linear Equation in two variables?
What is C in the general form of a line?
How the linear equations can be represented in matrix form?
Linear Equations were invented by Irish mathematician, physicist and astronomer Sir William Rowan Hamilton in 1843. Sir Hamilton have made the great contributions in field of mathematics.
He has made the special contribution in the algebra by inventing the linear equation. His main work was in the field of mathematical physics and thus the application of his work can be found in various fields in physics like quantum mechanics.
The word ‘Linear equation‘ is made up of two different words linear and equation. Linear means an algebraic expression having its degree (the highest power on the variable) as one. While, if any algebraic expression equated to zero or any other constant or expression becomes an equation.
Example
f(x) = 2x + 3y – 2z, g(x) = 2x – 1, h(x) = x – 3y + 2 are the examples of linear algebraic equation as the highest power of any variable is equals to one.
If the above linear expressions are equated to zero, then it becomes a linear equation.
that is,
f(x) = 0 ⇒ 2x + 3y – 2z = 0
also,
g(x) = 0 ⇒ 2x – 1 = 0
and
h(x) = 0 ⇒ x – 3y + 2
As explained above, linear equation is any algebraic equation having degree as one and is equated to zero.
In general, any linear equation having n different variables can be expressed as:
a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} + …………………….. + a_{n}x_{n} + c = 0
where, a_{1}, a_{2 },.. a_{n }are the coefficients of x_{1} , x_{2} , .. x_{n }respectively and c is any constant.
A linear equation in n variables can represents different graphical shapes.
x = 2 is a one dimension straight line, 2x + 3y = 1 is a two dimension straight line, but x – 7y + 10 = 0 represents the equation of the plane in three dimension and so on.
Below are few graphical examples of one, two and three-dimensional equation of line.
The general equation of a straight line in two variables can be represented as:
ax + by + c = 0
Where, a and b are the coefficients of x and y respectively and c is any arbitrary constant.
The reason, why we call the above linear equation in two variables as a straight line because when this equation is represented on the graph, it will be drawn as Straight Line.
Let us take one example and try to plot its graph.
y = 2x + 3
To plot the graph, we usually keep one variable on Right hand side and take other variables and constant on left hand side.
Now look for the vales of y for different vales of x.
For x = 0, we get y = 3
For x = 1, we get y = 5
For x = 2, we get y = 7
For x = 3, we get y = 9
For x = -1, we get y = 1
For x = -2, we get y = -1
We can also arrange these values in tabular form.
Now mark the above co-ordinates on a XY – plane. After joining these points, we will get the straight line as shown in the graph below:
The general form of a line is given by:
y = mx + c
Where, m is the slope and c represents the y – intercept of the equation of line.
y – intercept is an ordinate of a point on the y – axis where the line intersects with the y – axis. y – intercept can also be defined as the length of the line segment between the origin and the point where line intersects with y – axis.
The value of c can be positive, negative or even zero.
In y = mx + c, if c is positive that means for x = 0, the value of y = c which will always be positive.
Thus, the graph of a line at x = 0 will always be above the x – axis.
In y = mx + c, if c is negative that means for x = 0, the value of y = c which will again always be negative.
Thus, the graph of a line at x = 0 will always be below the x – axis.
In y = mx + c, if c is zero that means for x = 0, the value of y = c will also be zero. That means the point (0, 0) will always satisfy the equation.
Thus, the graph of a line will always pass through origin.
Matrix is an array of numbers represented as follows:
So, we can also represent a linear equation in the matrix form.
For ax + by + cz = 0, its matrix representation will be as
For solving the linear equations, we need as many number of different equations as the variables in the equation. For example, for two variable equations we need two equations. Similarly, for three variable equations, we need three equations and so on.
Thus, for solving the below equations:
2x + y + 3z = 4
x – y + 10z = -3
20x + 2y – 13z = 0
We can form a matrix as:
Then this can be solved using one of the matrix method to get the values of x, y and z.
More Readings
Signing up with Facebook allows you to connect with friends and classmates already using askIItians. It’s an easier way as well. “Relax, we won’t flood your facebook news feed!”
Post Question
Dear , Preparing for entrance exams? Register yourself for the free demo class from askiitians.
Examples on Distance between two Parallel Lines...
Centroid, Incentre, Circumcentre and Ex-centre of...
Concurrency of Straight Lines What do you mean by...
Angle Bisectors Table of Content What is an Angle...
Distance between two points Before we discuss the...
Distance between two Parallel Lines In this...
Angle between two Straight Lines Table of Content...
Position of two points with respect to a given...
Different Forms of Line Table of Content What do...
Area of a triangle Table of Content Area of a...
Examples based on Straight Line Straight Lines is...
Pair of Straight Lines The equation ax 2 + 2hxy +...
Examples on Angle between two Straight Lines...
Joint Equation of Pair of Lines Joint Equation of...
Family of Lines Table of Content What do you mean...
Combined Equations of Angle Bisectors of Lines...
Straight Line Any equation of first degree of the...
Various forms of the Equation of a Line Table of...
Slope of a Line Table of Content Define Straight...
Representation of Points in a Plane Table of...
Angle between Pair of Lines Straight lines is an...