Quick Revision Notes

Class: 9th Mathematics

Chapter: Linear Equations in Two Variables (दो चरों वाले रैखिक समीकरण)

### Concept 1: Definition of a Linear Equation in Two Variables

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

• Key Points:
  • Here, x and y are the two variables (चर).
  • a is the coefficient of x, b is the coefficient of y, and c is the constant term (अचर पद).
  • Example: 2x + 3y = 5 or 2x + 3y - 5 = 0 (where a = 2, b = 3, c = -5).

### Concept 2: Solution of a Linear Equation

• A solution to a linear equation in two variables is a pair of values, one for x and one for y (x = α, y = β), which makes the equation a true statement.
• A linear equation in two variables has infinitely many solutions (अपरिमित रूप से अनेक हल).
• Every solution of the linear equation represents a point on the graph of the equation.

### Concept 3: Graph of a Linear Equation in Two Variables

• The graph of every linear equation in two variables is a straight line (एक सरल रेखा).
• Steps to draw the graph:
  1. Express one variable in terms of the other (e.g., find y in terms of x).
  2. Put at least 2 or 3 different values for x and find the corresponding values of y to form a coordinate table.
  3. Plot these points on the Cartesian plane (कार्तीय तल).
  4. Join the points with a line.

### Concept 4: Equations of Lines Parallel to the X-axis and Y-axis

• Equation of the X-axis: y = 0
• Equation of the Y-axis: x = 0
• Equation of a line parallel to the X-axis: y = c (where c is a constant)
• Equation of a line parallel to the Y-axis: x = c (where c is a constant)

### Important Formulas & Tips for MP Board Exam

1. Standard Form: ax + by + c = 0

2. Converting to Standard Form Example:

   • Given: 2x = 3y
   • Standard form: 2x - 3y + 0 = 0 (Here a = 2, b = -3, c = 0)

3. Checking if a point (x, y) is a solution: Substitute the given coordinates into the L.H.S. of the equation. If L.H.S. = R.H.S., then it is a solution.

4. Number of Solutions:

   • Linear equation in one variable has a unique (one) solution.
   • Linear equation in two variables has infinitely many solutions.