Linear Equations Lecture Notes

Introduction

Linear equations are foundational concepts. They form the basis for solving more complex problems in mathematics.
Many students fear linear equations due to misunderstandings around foundational concepts.
Students should understand the basic principles and practice consistently.

Variables (">वेरियेबल्स<"): Represent unknown quantities (e.g., x, y, z). They change based on conditions or equations.
Constants (">कांस्टेंटस<"): Fixed values that do not change (e.g., 100, -3, etc.).
Terms (">टर्म्स<"): Combinations of variables and constants through multiplication (e.g., 3x, 2a²).
Expressions (">एक्सप्रेशंस<"): Combinations of terms connected by addition or subtraction (e.g., 2x + 5).
Equations: Statements that equate two expressions using an equals sign (e.g., 2x + 3 = 7).

Linear Equations in One Variable: Equations involving only one variable and the variable is raised to the power of one (e.g., 2x + 3 = 7).
Solution: The value of the variable that satisfies the equation.
Property of Equations: Whatever operation is performed on one side of the equation must also be performed on the other side to maintain equality.

Isolate the Variable: Move all terms containing the variable to one side and constants to the other side of the equation.
Simplification: Combine like terms and simplification steps to solve for the variable.
Verification: Substitute the solution back into the original equation to ensure it is correct.

Equation: 2x + 3 = 7
- Subtract 3 from both sides: 2x = 4
- Divide both sides by 2: x = 2
Equation: 3x - 5 = 1
- Add 5 to both sides: 3x = 6
- Divide both sides by 3: x = 2

Approach for Word Problems:
1. Carefully read the problem multiple times.
2. Translate the word problem into one or more equations.
3. Solve the equations for the unknowns.
Example: *