4.3 Solution of a Linear Equation

Description

Quick Overview

This section explores the nature of solutions for linear equations in two variables, focusing on the infinite solutions that can be derived from such equations.

Standard

In this section, we carefully analyze the solutions of linear equations involving two variables, highlighting that each equation can yield infinitely many pairs fitting the equation. We explore various methods of finding solutions through examples and exercises, reinforcing the concept that each solution corresponds to a point on the Cartesian plane.

Detailed

Solution of a Linear Equation

This section discusses the solutions to linear equations in two variables. Unlike linear equations in one variable that possess a unique solution, a linear equation in two variables provides us not just with one solution, but rather infinitely many pairs of values satisfying the equation form.

For example, the equation
$$2x + 3y = 12$$ is examined, demonstrating how different pairs of $(x, y)$ sets, such as (3, 2) and (0, 4), fulfill the equation's requirements. Students learn to verify candidates' solutions through substitution and are encouraged to explore choosing values for one variable to find corresponding values for the other variable.

The section includes practical examples to derive multiple solutions from a single linear equation, explaining methods like setting one variable to zero to find intercepts on the Cartesian plane. Through engaging exercises and examples, learners understand that each valid solution corresponds to a point on the linear equation's graph, revealing the infinite nature of solutions in this context.

Example : Find four different solutions of the equation \(x + 3y = 12\).
Solution: By inspection, \(x = 6, y = 2\) is a solution because for \(x = 6, y = 2\)
\[ 6 + 3 \times 2 = 12 \]
Now, let us choose \(x = 0\). With this value of \(x\), the equation reduces to \(3y = 12\) which has the unique solution \(y = 4\) so, \(x = 0, y = 4\) is also a solution of the given equation.
Similarly, taking \(y = 0\), the given equation reduces to \(x = 12\). So, \(x = 12, y = 0\) is a solution of \(x + 3y = 12\). Finally, let us take \(y = 1\). The given equation now reduces to \(x + 3 = 12\), whose solution is \(x = 9\). Therefore, \((9, 1), (6, 2), (0, 4), (12, 0)\) is also a solution of the given equation. So four of the infinitely many solutions of the given equation are:
\[(6, 2), (0, 4), (12, 0), (9, 1)\]

Key Concepts

  • Linear Equation: An equation of the first degree involving two variables.

  • Ordered Pair: A solution represented as (x, y) satisfying the equation.

  • Infinite Solutions: The nature of linear equations in two variables leading to endless solutions.

  • Substitution Method: A technique used to determine solutions by replacing a variable.

Memory Aids

🎵 Rhymes Time

  • For every linear line you see, pairs of values must come free.

📖 Fascinating Stories

  • Alice and Bob share a total of 100 apples. For every apple Alice gets, Bob gets two. They can trade any way they like, but the total remains constant!

🧠 Other Memory Gems

  • SPLIT: Substitute, Pair, Linear, Infinite, Test if valid.

🎯 Super Acronyms

SPL

  • Solutions
  • Pairs
  • Lines.

Examples

  • {'example': 'Find different solutions for the equation 2x + 3y = 12.', 'solution': 'Solutions include (3, 2), (0, 4), (6, 0), (2, 4/3), etc.'}

  • {'example': 'Verify if (2, 1) is a solution for 2x + 3y = 12.', 'solution': 'Substituting gives 2(2) + 3(1) = 4 + 3 = 7, not a solution.'}

Glossary of Terms

  • Term: Linear Equation

    Definition:

    An equation involving two variables that can be written in the form ax + by + c = 0.

  • Term: Ordered Pair

    Definition:

    A pair of numbers (x, y) that represents a solution to a linear equation.

  • Term: Infinite Solutions

    Definition:

    A scenario where a linear equation in two variables can yield endless valid pairs (x, y).

  • Term: Substitution

    Definition:

    A method of replacing a variable with a number or another expression in order to solve an equation.