Quick Revision Notes for Class 10 Maths Chapter 3 Linear Equations in Two Variables

Here we are providing Quick Revision Notes, Common mistakes we do while solving problems and Types of questions asked in Linear Equations for Class 10 Maths. Students are advised to check out these points to score better in exams.

Quick Revision Notes for Class 10 Maths Chapter 3 Linear Equations in Two Variables

• Two linear equations in the same two variables are called a pair of linear equations in two variables.

• The general form of a pair of linear equations is a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ =0; where a₁, a₂, b₁, b₂, c₁, c₂ are real numbers, such that a₁²+b₁²≠0, a₂²+b₂²≠0

• A pair of linear equations in two variables can be represented and solved by the (i) Graphical Method (ii) Algebraic Method
  Graphical Method: The graph of a pair of linear equations in two variables is represented by two lines.
  (i) If the lines intersect at a point, then the point gives the unique solution of the two equations. The pair of equations is consistent.
  (ii) If the lines are parallel, the pair of lines of equations has no solution. The pair of equations are inconsistent.
  (iii) If the lines coincide, then there are infinitely many solutions. The pair of equations is consistent.
  Algebraic Method: The following are the algebraic methods:
  (i) Substitution Method
  (ii) Elimination Method
• If a pair of linear equations is given by a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, then the following situations can arise:
  (i)a₁/a₂ ≠ b₁/b₂ ⇒ Equations are consistent, Intersecting Lines, Exactly one solution (Unique)
  (ii) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ⇒ Equations are inconsistent, Parallel lines, No solution.
  (iii) a₁/a₂ = b₁/b₂ = c₁/c₂ ⇒ Equations are dependent and are consistent, Coincident lines, Infinitely many solutions.
• There are many situations which can be mathematically represented by two equations that are not linear to start with. But, they can be reduced to a pair of linear equations.

Common mistakes we do while solving problems.

We are listing common mistakes we do while solving Linear equation problems. Take care of these points to avoid these common errors and score better marks in Maths.

(1) Error: Incorrectly transposing the terms.
Correction: Be careful about sign of terms while transposing.

(2) Error: Making mistake in the problem of upstream and downstream.

For example, let the speed of the boat in still water be x km/h and speed of the stream be y km/h.
Then, the speed of the boat downstream = (x – y) km/h and the speed of the boat upstream = (x + y) km/h
Correction: The speed of the boat downstream= (x + y) km / h and the speed of the boat upstream = (x – y) km / h

(3) Error: Writing incorrectly that if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ , there are infinitely many solutions.
Correction:  If a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ⇒ Equations are inconsistent and has no solutions.

Types of Questions Asked:

1. To find the value of unknown for which the given pair of linear equations has unique solution, infinitely many solutions or no solutions.
2. To find out whether the lines representing the given pairs of linear equations intersect at a point, are parallel or coincident by comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂ 
3. To draw the graphs of given equations and determine the coordinates of the vertices of the triangle formed by the lines and the x-axis.
4. To solve the given pair of linear equations by the substitution method and elimination methods.
5. To solve the given pair of equations by reducing them to a pair of linear equations.
6. To formulate the given problem as a pair of linear equations and hence find its solution.

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