Recording-2025-01-16 1.1pp linear eq

Understanding Linear Equations

• Linear equations are defined by the following characteristics:

  • They must have a variable raised to the first power.

  • The variable must not be under any radicals or in the denominator.

  • Example: equations like (x + 2 = 5) or (6x - 12 = 2) are linear.

Non-Linear Equations

• Equations that do not meet the criteria for linearity:

  • These include variables that are under square roots, such as (\sqrt{x} = 4).

  • Although we will not cover these today, they can still be solved using various methods.

Solving Linear Equations

• To solve linear equations, we apply:

  • Addition and Multiplication Properties of Equality: This involves doing the same operation on both sides of the equation to maintain balance.

• Example process when simplifying:

  1. Simplify both sides of the equation if possible.

  2. Identify which terms can be combined.

     • For example, in the equation (6x - 12 = 7 - 5):

       • Left Side: (6x - 12) (already simplified, no like terms)

       • Right Side: (7 - 5) can be simplified to (2).

Steps to Isolate the Variable

• After simplification, we proceed with operations to isolate the variable:

  • We can choose to:

    • Subtract or add values from both sides.

  • Example actions include:

    • Subtracting (6x) from both sides, adding (12) to both sides, etc.

Importance of Checking Solutions

• After obtaining a solution, it is a good habit to verify:

  • Checking the answer helps validate if it satisfies the original equation and can help catch mistakes.

  • While it may seem unnecessary in simple cases, developing this habit is crucial for more complex equations later on.