4.3.3 Examples of Solutions

Description

Quick Overview

This section explores the solutions to linear equations in two variables, emphasizing the idea that each equation typically has infinitely many solutions.

Standard

In this section, students learn that linear equations in two variables often have multiple solutions, represented as ordered pairs. They can substitute values for one variable to find corresponding values for another, enabling the identification of various solutions.

Detailed

Detailed Summary

In this section, we focus on the concept of solutions for linear equations in two variables. Unlike linear equations in one variable, which have a unique solution, linear equations in two variables can yield infinitely many solutions. For example, the equation \(2x + 3y = 12\) has multiple solutions like \((3, 2)\) and \((0, 4)\). To find more solutions, students can assign a specific value to \(x\) or \(y\) and solve for the other variable, illustrating the rich set of points that satisfy the given equation.

Through examples like \(x + 2y = 6\), students are guided to find several valid ordered pairs, emphasizing that one can generate an endless list of solutions. The importance of ordered pairs, the role of substitution, and understanding the graphical representation of these solutions on the Cartesian plane are vital concepts introduced in this section.

Key Concepts

  • Ordered pairs represent solutions to linear equations in two variables.

  • A linear equation can have infinitely many solutions.

Memory Aids

🎵 Rhymes Time

  • Solutions are pairs, like x and y, plug in the numbers, watch them fly!

📖 Fascinating Stories

  • Imagine x is a path, y is a tree. Together they create a solution, just wait and see!

🧠 Other Memory Gems

  • X+Y=Pairs for a linear equation's cares!

🎯 Super Acronyms

SOLUTION

  • Set values
  • Learn if they work
  • Obtain
  • True or false
  • Identify Oh! Note!

Examples

  • {'example': 'Find four solutions for the equation \(x + 2y = 6\).', 'solution': 'The solutions can be derived as follows:\n1. Let \(x = 0\), then \(2y = 6 \Rightarrow y = 3\), giving us \((0, 3)\).\n2. Let \(y = 0\), then \(x = 6\), yielding \((6, 0)\).\n3. Let \(x = 2\), then \(2y = 4 \Rightarrow y = 2\), thus \((2, 2)\).\n4. Let \(y = 1\), then \(x + 2 = 6 \Rightarrow x = 4\), giving \((4, 1)\).'}

  • {'example': 'Verify if the pairs \((0, 2)\) and \((4, 0)\) are solutions for the equation \(x - 2y = 4\).', 'solution': 'For \((0, 2)\): \(0 - 2(2) = -4 \neq 4\) (not a solution).\nFor \((4, 0)\): \(4 - 2(0) = 4 \) (is a solution).'}

Glossary of Terms

  • Term: Linear Equation

    Definition:

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

  • Term: Ordered Pair

    Definition:

    A pair of numbers \((x, y)\) that represents a point on the Cartesian plane.

  • Term: Infinitely Many Solutions

    Definition:

    Refers to the behavior of linear equations in two variables, where numerous pairs \((x, y)\) can satisfy the equation.