Linear Equations in two Variables

Linear Equations in two Variables


What is linear equation?

 

The definition of linear equation is: “The first degree equations in one, two, or three variables are called Linear Equations.”

So, the equations of the form: ax + b = 0, ax + by + c = 0 and ax + by + cz + d = 0 are linear equations in one, two and three variables x, y and z where a, b, c and d are real numbers and not equal to zero at a time.

 

x = 1, 2x + 3 = 0, x + 4y = 7, 5x – 2y = 3, x + 3y – 4z = 1 etc. are some examples of linear equations.

 

 

What does a Linear equation do?

 

In coordinate geometry,

-     Linear equations in one variable represents the straight lines parallel to the axes on a coordinate plane.

-     Linear equations in two variables represent the oblique straight lines on a coordinate plane.

-     Linear equations in three variables always represent a plane on a three-dimensional space.

 

  

Linear Equations in two Variables

 

Linear equation in two variables x and y is an equation of the form ax + by + c = 0, where a, b and c are real numbers and both a and b are not equal to 0 at a time.

 

2x + 3y = 6 and x + 4y = 7 are some examples of linear equation in two variables. These equation can also written in the form 2x + 3y – 6 = 0,  x + 4y – 7 = 0 respectively.

 

Note: Equations of the type ax+b=0 are also examples of linear equations in two variables because they can be expressed as ax+0.y+b=0. For example, 4-3x=0 can be written as -3x+0.y+4=0

 

 

Solving Linear Equations

 

Solving linear a equation ax + by + c = 0 is to find the points (x, y) or pair of values of x and y which satisfy the given linear equation. For example the point (2, 3) is the solution of the linear equation 2x + 3y – 13 = 0. Because for the value of x = 2 and y = 3 satisfy the equation 2x + 3y – 13 = 0,

 

i.e      2x2 + 3x3 – 13 = 0

or,     4 + 9 – 13 = 0

or,     13 – 13 = 0

or,     0 = 0 (True)

 

 

How to Solve Linear Equation?

 

To solve a linear equation in two variables x and y, we express the value of any one variable in terms of another variable. And then, by choosing some values for the second variable, we can find the corresponding values of the first variable.

 

For example, look at the following solutions of linear equations.

 

Example 1: Find the solutions of the linear equation y – 3x = 4.

 

Solution: Here,

Finding the value of y in terms of x,

y – 3x = 4

or,     y = 4 + 3x

 

Now, by choosing some values of x we can find the corresponding values of y as given in the table below.


Table,

Example 1: Table

The the solutions of linear equation y – 3x = 4 are: (0, 4), (1, 7), (-1, 1), (2, 10) and (-2, -2)

 

 

Example 2: Find the solutions of the linear equation 3x + 4y = 10.

 

Solution: Here,

Finding the value of x in terms of y,

3x + 4y = 10

or,     3x = 10 – 4y

or,     x = (10 – 4y)/3

 

Now, by choosing some values of y we can find the corresponding values of x as given in the table below.


Table,

Example 2: Table

The the solutions of the linear equation 3x + 4y = 10 are: (2, 1), (-2, 4), (-6, 7), (-10, 10) and (-14, 13)

 

Note: A linear equation in two variables has infinitely many solutions.

 

 

Graphing Linear Equations

 

By plotting the solution points of a linear equation on a coordinate plane and joining them, we will get a straight line on the graph, which is called the Graph of Linear Equation.

 

Some of the special cases:


1.     x=0 is the equation of y-axis and y=0 is the equation of the x-axis.

2.     The graph of x=a is a straight line parallel to the y-axis.

3.     The graph of y=b is a straight line parallel to the x-axis.

4.     The graph of the equation in the form y=mx is a line that always passes through the origin.

 

For example, look at the following solutions of linear equations and their graphs.


 

Example 3: Draw the graph of x + y = 7.

 

Solution: Here,

x + y = 7

or,     x = 7 – y

 

Table,

Example 3: Table

 Three solutions of the given linear equation x + y = 7 are (4, 3), (3, 4) and (2, 5). 

 

Now, plotting these points on a graph, and joining them by a straight line we get following graph of linear equation.


Graph of Linear Equation x + y = 7



Example 4: Draw the graph of 2x – y = 1.

 

Solution: Here,

2x – y =1

or,     2x = 1 + y

or,     x = (1 + y)/2

 

Table,

Example 4: Table

 Three solutions of the given linear equation 2x – y = 1 are (1, 1), (2, 3) and (3, 5). 

 

Now, plotting these points on a graph, and joining them by a straight line we get following graph of linear equation.


Graph of Linear Equation 2x - y = 1


Example 5: Draw the graph of 2x + 3y = 6.

 

Solution: Here,

2x + 3y = 6

or,     2x = 6 – 3y

or,     x = (6 – 3y)/2

 

Table,

Example 5: Table

 Three solutions of the given linear equation 2x + 3y = 6 are (3, 0), (0, 2) and (-3, 4). 

 

Now, plotting these points on a graph, and joining them by a straight line we get following graph of linear equation.


Graph of Linear Equation 2x + 3y = 6

 

If you have any questions or problems regarding the Linear Equations in Two Variables, you can ask here, in the comment section below.

 

 

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