Definitions [1]
Definition: Linear Equations in Two Variables
An equation which contains two variables and the degree of each term containing a variable is one is called a linear equation in two variables.
General Form:
ax + by + c = 0
In simple words, all three must hold:
- It has two variables (usually x and y).
- Every variable term has power 1 (no x², xy, 1/x, √x).
- It can be written as ax + by + c = 0, with a and b not both zero.
Key Points
Key Points: Elimination Method
Steps:
-
Multiply one or both equations
-
Add or subtract
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Solve
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Substitute
| Case | Result While Solving | Type of Statement | Nature of Pair |
|---|---|---|---|
| Unique solution | You get a specific value of x (or y) | Valid numerical value | Consistent |
| Infinitely many solutions | You get a true statement (e.g. 18 = 18) | Always true | Dependent |
| No solution | You get a false statement (e.g. 0 = 9) | Contradiction | Inconsistent |
Key Points: Substitution Method
Steps:
-
Express one variable in terms of the other
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Substitute
-
Solve
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Substitute back
| Case | What You Get While Solving | Type of Statement | Nature of Equations |
|---|---|---|---|
| Unique solution | Specific values of x and y | Valid numerical values | Consistent |
| Infinitely many solutions | A true statement like 18 = 18 | Always true | Dependent |
| No solution | A false statement like 0 = 5 | Contradiction | Inconsistent |
Key Points: Foundation of Linear Equations in Two Variables
- General form: ax + by + c = 0, a and b not both zero.
- Every variable term has degree 1.
- The graph is a straight line.
- To reach the general form, shift all terms to one side.
Concepts [6]
- Pair of Linear Equations in Two Variables
- Algebraic Methods of Solving a Pair of Linear Equations
- Elimination Method
- Substitution Method
- Foundation of Linear Equations in Two Variables
- Application of Linear Equation in Two Variables
