Lecture Notes: Linear Equations

Definition

• An equation is formed when two expressions are separated by an equality sign.
• The equality sign indicates that the expressions on either side are equal.
• Example: 5 + 2 = 7.

Key Components

• Variables/Unknowns: Represented by letters, commonly x and y.
• Constant: A number on its own.
• Coefficient: A number that multiplies a variable (e.g., in 6x, 6 is the coefficient).

Standard Form

• Linear equation in one variable: ( ax + b = c ), where ( a, b, ) and ( c ) are real numbers and ( a \neq 0 ).
• Example: 3x + 4 = 13.
• The exponent of the variable is always 1.

Solution

• The solution is the value that makes the equation true when substituted for the variable.
• Example: x = 4 is a solution of x + 5 = 9 because 4 + 5 = 9.

Solving Linear Equations

• Objective: Isolate the variable on one side of the equation.
• Example 1: Solve 2x + 3 = 11
  • Subtraction: 2x + 3 = 11 implies 2x = 11 - 3
  • Division: 2x = 8 implies x = 8/2 = 4
  • Solution: x = 4
• Example 2: Solve x = x + 2
  • Simplification: x - x = 2 implies 0 = 2 (false)
  • Conclusion: No solution

Different Types of Solutions

• One Solution: A specific value makes the equation true.
• No Solution: The equation results in a false statement (e.g., 0 = 2).
• Infinitely Many Solutions: The equation results in a true statement for all values of the variable (e.g., 0 = 0).
  • Example 3: 2x + 4 = x + 4 + x
    • Simplification leads to 0 = 0
    • Conclusion: Infinitely many solutions

Conclusion

• A linear equation in one variable may have one solution, infinitely many solutions, or no solution.

Additional Resources

• For more exercises: Visit MOOC NIOS
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