Linear Equations in One Variable

2 Linear Equations in One Variable

Description

Quick Overview

This section introduces linear equations in one variable, explaining their characteristics and methods for solving them.

Standard

In this section, students revisit algebraic expressions and equations, focusing specifically on linear equations in one variable. The section outlines how to identify linear expressions, demonstrates solving equations involving variables on both sides, and emphasizes techniques for simplifying equations to find solutions.

Detailed

Detailed Summary

In this section, we explore linear equations in one variable, which are characterized by their linear nature, meaning the variable involved has an exponent of one. We differentiate between algebraic expressions (like 5x and 2x - 3) and equations (where an equality sign = is present).

Key Aspects Covered:
- Definitions: Understanding what constitutes an algebraic equation and the significance of the left-hand side (LHS) and right-hand side (RHS) in an equation.
- Solving Techniques: Methods to solve equations with variables on both sides by manipulating expressions while maintaining equality. For instance, in the equation 2x - 3 = x + 2, students learn to isolate the variable through transposition and addition or subtraction of terms.
- Examples: Practical examples illustrate the solving process in detail, demonstrating essential algebraic manipulations.
- Equations with More Complexity: We also discuss simplifying more complicated equations, such as those containing fractions or requiring the use of the least common multiple (LCM) for solving denominators.

By mastering these techniques, students will be equipped to tackle various mathematical problems involving linear equations effectively.

Key Concepts

  • Linear Equation: An equation where the highest power of the variable is one.

  • Variables: Symbols representing unknown values in equations.

  • LHS and RHS: The left and right sides of an equation that must be equal.

Memory Aids

🎵 Rhymes Time

  • Linear equations might sound a chore, but with 'solve and balance', you'll never bore!

📖 Fascinating Stories

  • Once upon a time, in Equatia, there lived a wise old mathematician who said, 'To solve an equation, keep both sides equal - treat them as best friends.'

🧠 Other Memory Gems

  • Remember 'SOLVE': Separate, Operate, Leave variable alone, Verify solution, Eliminate mistakes.

🎯 Super Acronyms

Use 'L.E.S.S.' to remember - Linear equations, Equal sign, Simplifying expressions, Solve for variable.

Examples

  • To solve the equation 2x + 3 = 7, we subtract 3 from both sides, resulting in 2x = 4, and then divide by 2 to get x = 2.

  • In 3(x - 2) = 9, we first divide by 3 to get x - 2 = 3, and then add 2 to solve for x = 5.

Glossary of Terms

  • Term: Linear Equation

    Definition:

    An equation involving a variable raised to the first power.

  • Term: Variable

    Definition:

    A symbol used to represent an unknown value in an equation.

  • Term: Expression

    Definition:

    A combination of numbers, variables, and operations without an equality sign.

  • Term: LeftHand Side (LHS)

    Definition:

    The expression on the left side of the equality sign in an equation.

  • Term: RightHand Side (RHS)

    Definition:

    The expression on the right side of the equality sign in an equation.