4.3.1 Unique Solution for Linear Equations

Description

Quick Overview

This section discusses how linear equations in two variables can have infinitely many solutions, contrasting them with unique solutions in one-variable equations.

Standard

In this section, students learn that linear equations in two variables do not have a unique solution but rather an infinite number of solutions represented as pairs of values (x, y). The section emphasizes the identification and verification of solutions through examples.

Detailed

Unique Solutions in Linear Equations

In this section, we explore the characteristics of linear equations with two variables, emphasizing that unlike linear equations in one variable, which have a unique solution, linear equations in two variables can have infinitely many solutions. A solution to such an equation is represented as an ordered pair (x, y) that satisfies the equation. For example, the equation 2x + 3y = 12 has solutions such as (3, 2), (0, 4), and even (6, 0). This is because by choosing different values for either variable, corresponding values can be calculated for the other variable.

Further, the section illustrates finding solutions through various approaches, including substituting values directly and validating whether given pairs are indeed solutions to the equations. A focus is laid on practical exercises and examples demonstrating how a single equation can yield multiple valid solutions, reflecting the richness and flexibility in handling two-variable linear equations.

Key Concepts

  • Linear equations in two variables have infinitely many solutions.

  • Solutions are represented as ordered pairs (x, y).

  • Substituting values for one variable allows finding multiple corresponding values for the other variable.

Memory Aids

🎵 Rhymes Time

  • With x and y in play, many solutions come our way.

📖 Fascinating Stories

  • Imagine you have a treasure map with many paths. Each path represents a solution (x, y) in two-variable equations. Each correct pair leads to the treasure.

🧠 Other Memory Gems

  • S.O.L.V.E.: Start with one variable, Output the other, List several pairs, Verify them!

🎯 Super Acronyms

P.A.I.R.S.

  • Pick
  • Assign
  • Input
  • Reveal Solutions.

Examples

  • For the equation 2x + 3y = 12, valid solutions include (3, 2), (0, 4), and (6, 0).

  • In the equation x + 2y = 6, possible solutions are found by letting x = 0 (resulting in (6, 0)) and y = 0 (resulting in (0, 3)).

Glossary of Terms

  • Term: Linear Equation

    Definition:

    An equation that can be plotted as a straight line on a graph.

  • Term: Ordered Pair

    Definition:

    A pair of values (x, y) that satisfies the given equation.

  • Term: Infinitely Many Solutions

    Definition:

    The situation where an equation has unlimited solutions usually represented as pairs (x, y).