CBSE Linear Equations Subject Notes
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Linear Equations
Introduction to Linear Equations in Two Variables
Linear equation in two variable: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero (a^{2} + b^{2} ≠ 0), is called a linear equation in two variables x and y.
Solution of a linear equation in two variables: Every solution of the equation is a point on the line representing it.
Each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.
General form of pair of linear equations in two variables:
The general form for a pair of linear equations in two variables x and y is
a_{1}x + b_{1}y + c_{1} = 0 and
a_{2}x + b_{2}y + c_{2} = 0,
Where a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are all real numbers and a_{1}^{2} + b_{1}^{2} ≠ 0, a_{2}^{2} + b_{2}^{2} ≠ 0.
Geometrical representation of pair of linear equations in two variables
The geometrical representation of a linear equation in two variables is a straight line.
Pair of linear equations in two variables:
If we have two linear equations in two variables in a plane, and we draw lines representing the equations, then:
Condition 

Result 
Lines intersecting at a single point 
=> 
The pair of equations has a unique solution. The pair of linear equations is consistent 
Lines parallel to each other 
=> 
No solutions. The pair of linear equations is inconsistent. 
Coincident lines 
=> 
Infinite number of solutions. The pair of linear equations is consistent and dependent. 
Algebraic interpretation of pair of linear equations in two variables
The pair of linear equations represented by these lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0
 If then the pair of linear equations has exactly one solution.
 If then the pair of linear equations has infinitely many solutions.
 If then the pair of linear equations has no solution.
S. No. 
Pair of lines 
a_{1}/a_{2} 
b_{1}/b_{2} 
c_{1}/c_{2} 
Compare the ratios 
Graphical representation 
Algebraic interpretation 
1 
a_{1}x + b_{1}y + c_{1} = 0
a_{2}x + b_{2}y + c_{2} = 0 
a_{1}/a_{2} 
b_{1}/ b_{2} 
c_{1}/c_{2} 
a_{1} / a_{2} ≠ b_{1} / b_{2} 
Intersecting lines 
Unique solution (Exactly one solution) 
2 
a_{1}x + b_{1}y + c_{1} = 0
a_{2}x + b_{2}y + c_{2} = 0 
a_{1}/ _{2} 
b_{1}/b_{2} 
c_{1}/c_{2} 
a_{1}/ a_{2} = b_{1} / b_{2} = c_{1} / c_{2} 
Coincident lines 
Infinitely many solutions 
3 
a_{1}x + b_{1}y + c_{1} = 0
a_{2}x + b_{2}y + c_{2} = 0 
a_{1}/a_{2} 
b_{1}/ b_{2} 
c_{1} / c_{2} 
a_{1} / a_{2}= b_{1} / b_{2} ≠ c_{1} / c_{2} 
Parallel lines 
No solution 
(a) Substitution method: Following are the steps to solve the pair of linear equations by substitution method:
a_{1}x + b_{1}y + c_{1} = 0 … (i) and
a_{2}x + b_{2}y + c_{2} = 0 … (ii)
Step 1: We pick either of the equations and write one variable in terms of the other
y = a_{1} / b_{1} x  c_{1} / b_{1}… (iii)
Step 2: Substitute the value of x in equation (i) from equation (iii) obtained in step 1.
Step 3: Substituting this value of y in equation (iii) obtained in step 1, we get the values of x and y.
