Notes of Linear Equations in Two Variables

Framing a Linear Equation

Linear equation in one variable

An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables

Linear equations in one variable, of the type ax + b = 0, can also expressed as a linear equations in two variables. Since, ax + b = 0 ⇒ ax + 0.y + b = 0. 

A solution of a linear equation in two variables is a pair of values, one for x and one for y, which satisfy the equation. 

  • The solution of a linear equation is not effected when 
  • The same number is added or subtracted from both the sides of an equation. 
  • Multiplying or dividing both the sides of the equation by the same non-zero number   

When an equation has only one variable of degree one, then that equation is known as linear equation in one variable.

  • Standard form: ax+b=0, where a and b ϵ R & a ≠ 0
  • Examples of linear equation in one variable are :
    • 3x-9 = 0
    • 2t = 5

Linear equation in 2 variables

When an equation has two variables both of degree one, then that equation is known as linear equation in two variables.

Standard form: ax+by+c=0, where a,b,c ϵ R & a,b ≠ 0 Examples of linear equations in two variables are:

  • 7x+y=8
  • 6p-4q+12=0

Examples of a Linear Equations

Solution of linear equation in 2 variables

A linear equation in two variables has infinitely many solutions

Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line. 

A linear equation in two variables is represented geometrically by a straight line whose points make up the collection of solutions of the equation. This is called the graph of the linear equation. 

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

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

For example, the graph of the equation x = 5 is as follows: 

1

A linear equation in two variables has a pair of numbers that can satisfy the equations. This pair of numbers is called as the solution of the linear equation in two variables.

  • The solution can be found by assuming the value of one of the variable and then proceed to find the other solution.
  • There are infinitely many solutions for a single linear equation in two variables.

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

For example, the graph of the equation y = 5 is as follows: 

2

An equation of the type y = mx represents a line passing through the origin, where m is a real number. For example, the graph of the equation y = 2x is as follows: 

3

Graph of a Linear Equation

Graphical representation of a linear equation in 2 variables

  • Any linear equation in the standard form ax+by+c=0 has a pair of solutions (x,y), that can be represented in the coordinate plane.
  • When an equation is represented graphically, it is a straight line that may or may not cut the coordinate axes.
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Solutions of Linear equation in 2 variables on a graph

  • A linear equation ax+by+c=0 is represented graphically as a straight line.
  • Every point on the line is a solution for the linear equation.
  • Every solution of the linear equation is a point on the line. Lines passing through origin
  • Certain linear equations exist such that their solution is (0,0). Such equations when represented graphically pass through the origin. 
  • The coordinate axes x-axis and y-axis can be represented as y=0 and x=0 respectively. Lines parallel to coordinate axes
  • Linear equations of the form y=a, when represented graphically are lines parallel to the x-axis and a is the y-coordinate of the points in that line.
  • Linear equations of the form x=a, when represented graphically are lines parallel to the y-axis and a is the x-coordinate of the points in that line.