4 LINEAR EQUATIONS IN TWO VARIABLES

Description

Quick Overview

This section introduces linear equations in two variables, exploring their characteristics, solutions, and representation on the Cartesian plane.

Standard

The section explains the concept of linear equations in two variables, providing definitions, examples, and showing how to represent solutions graphically. It emphasizes the infinite nature of solutions and offers insights into forming equations based on real-life scenarios.

Detailed

In this section, linear equations in two variables are defined as equations that can be expressed in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. The section starts by recalling previous knowledge of linear equations in one variable and extends it to two variables, focusing on questions such as whether solutions exist and their uniqueness.

Examples demonstrate how to convert various equations into the standard form and identify their coefficients. The section highlights that a linear equation in two variables has infinitely many solutions, where each solution corresponds to a point on the Cartesian plane. Additionally, exercises encourage students to practice identifying and writing linear equations, while examples illustrate the method of finding solutions by setting values for one variable and solving for the other. Overall, this section lays the foundation for understanding linear equations and their representation, preparing students for further exploration in the chapter.

Key Concepts

  • Linear equations can be expressed in the form ax + by + c = 0.

  • Linear equations in two variables have infinitely many solutions.

  • Each solution of a linear equation corresponds to a point on the Cartesian plane.

Memory Aids

🎵 Rhymes Time

  • If x and y are in one, a line is where they run; the points they make are always great, come and solve; don’t wait!

📖 Fascinating Stories

  • Imagine two friends at a park, sharing a total of 20 candies. They don't know how many each has, but whatever they take keeps the total the same. This is like a linear equation in two variables - it's their mystery to solve together!

🧠 Other Memory Gems

  • Use 'SOLVE' - Substitute numbers, Observe results, Log the pairs, Verify solutions, everywhere on the graph!

🎯 Super Acronyms

Remember 'LINE' for Linear Is Not Ending, emphasizing the infinite solutions of linear equations!

Examples

  • {'example': 'Write each of the following equations in the form ax + by + c = 0 and indicate the values of a, b and c.', 'solution': '(i) For 2x + 3y = 4.37, we get 2x + 3y - 4.37 = 0, where a = 2, b = 3, c = -4.37.'}

  • {'example': 'Find four different solutions for the equation x + 2y = 6.', 'solution': 'We can find solutions by choosing values: (2, 2), (0, 3), (6, 0), and (4, 1) are all valid solutions.'}

Glossary of Terms

  • Term: Linear Equation

    Definition:

    An equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and not both a and b are zero.

  • Term: Solution

    Definition:

    A pair of values (x, y) that satisfy a given linear equation.

  • Term: Cartesian Plane

    Definition:

    A two-dimensional plane formed by the intersection of a horizontal x-axis and a vertical y-axis.

  • Term: Ordered Pair

    Definition:

    A pair of numbers (x, y) used to represent a point on the Cartesian plane.