4.2 Linear Equations

Description

Quick Overview

This section introduces linear equations in two variables, emphasizing their structure, solutions, and various forms.

Standard

In this section, we explore linear equations in two variables, discussing their general form (ax + by + c = 0), solution sets, and examples. The section highlights the characteristics of such equations and their representation on the Cartesian plane.

Detailed

Linear Equations in Two Variables

This section delves into the concept of linear equations in two variables. A linear equation can be generally expressed in the form ax + by + c = 0, where a and b are not both zero. It emphasizes the relationship between the variables and introduces the idea of solutions, which are ordered pairs (x, y). Unlike linear equations in one variable, linear equations in two variables can generate infinitely many solutions, as demonstrated through various examples. Additionally, the section provides exercises aimed at transforming equations into the standard form and encourages problem-solving as a method to understand the relationships between variables in real-life contexts.

Example 2 : Write each of the following as an equation in two variables:

(i) \( x = 7 \)
(ii) \( y = 3 \)
(iii) \( 5x = 15 \)
(iv) \( 4y = 20 \)

Solution :
(i) \( x = 7 \) can be written as \( 1 \cdot x + 0 \cdot y = 7 \), \( 0 \cdot x + 1 \cdot y + 7 = 0 \).

(ii) \( y = 3 \) can be written as \( 0 \cdot x + 1 \cdot y - 3 = 0 \).

(iii) \( 5x = 15 \) can be written as \( 5 \cdot x - 15 = 0 \).

(iv) \( 4y = 20 \) can be written as \( 0 \cdot x + 4 \cdot y - 20 = 0 \).

Key Concepts

  • Linear Equation: A fundamental mathematical expression involving variables that can be represented graphically as a straight line.

  • Infinitely Many Solutions: Each linear equation in two variables has an infinite set of solutions, depicted as points on a line in a Cartesian plane.

  • Standard Form: The standard form of a linear equation, ax + by + c = 0, helps identify the coefficients and constant clearly.

Memory Aids

🎵 Rhymes Time

  • When x and y are combined, a line you will surely find.

📖 Fascinating Stories

  • Imagine a river, x and y are its banks. They flow together forming a linear path, with points that connect them.

🧠 Other Memory Gems

  • A: Always, B: Bring, C: Cookies - A = coefficients of x, B = coefficients of y, C = constant term.

🎯 Super Acronyms

LINEAR

  • L: = Line
  • I: = In
  • N: = Normal
  • E: = Equation
  • A: = All
  • R: = Relations.

Examples

  • Example 1: To convert 2x + 3y = 4.37 into standard form, write it as 2x + 3y - 4.37 = 0.

  • Example 2: If given x = 5 as a linear equation, it can be expressed in standard form as x + 0. y - 5 = 0.

Glossary of Terms

  • Term: Linear Equation

    Definition:

    An equation of the form ax + by + c = 0, where a and b are real numbers, and not both a and b are zero.

  • Term: Variable

    Definition:

    A symbol (often x or y) representing a number in equations.

  • Term: Solution Set

    Definition:

    The collection of all ordered pairs (x, y) that satisfy a given linear equation.

  • Term: Ordered Pair

    Definition:

    A pair of numbers (x, y) representing the solution to a linear equation.