Substitution Method

3.3.1 Substitution Method

Description

Quick Overview

The Substitution Method is an effective algebraic technique used to solve pairs of linear equations by expressing one variable in terms of the other and substituting it into the other equation.

Standard

In this section, we explore the Substitution Method, a systematic approach to solving pairs of linear equations. By rearranging one equation to isolate a variable and substituting that expression into the other equation, we can find the values of both variables. The section includes several examples and emphasizes the importance of verification after obtaining the solution.

Detailed

Substitution Method in Linear Equations

The Substitution Method is a powerful algebraic technique for solving a pair of linear equations. It allows one to find the values of variables by expressing one variable in terms of the other and then substituting that expression into the second equation. Here are the primary steps and components involved in this method:

Key Steps:

  1. Isolate a variable: Choose one of the two equations and solve for one variable in terms of the other. This is often easier if the equation is already in a form that allows easy isolation.
  2. Substitution: Substitute the expression obtained for one variable into the other equation. This will yield an equation with only one variable which can be solved easily.
  3. Finding both variables: Once one variable is solved, substitute this value back into one of the original equations to find the second variable.
  4. Verification: It is crucial to substitute both values back into the original equations to ensure they satisfy both equations.

Importance:

The Substitution Method not only provides a systematic way to solve linear equations but also enhances understanding of relationships between variables. By utilizing this method, learners gain familiarity in manipulating algebraic expressions and equations, setting a foundation for more complex algebraic concepts.

Key Concepts

  • Isolating a variable: A crucial step in the substitution method where one variable is expressed in terms of another.

  • Substituting values: Inserting the expression of the isolated variable into the other equation to solve for the remaining variable.

  • Verification: The process of checking if the obtained solutions satisfy the original equations.

Memory Aids

🎵 Rhymes Time

  • Substitution, it's no illusion, solve equations in a truer fashion.

📖 Fascinating Stories

  • Once there was a wise old owl who taught students to express their thoughts clearly. She always started with isolating a variable, then asked them to substitute their ideas into the world of equations.

🧠 Other Memory Gems

  • I.S.S.V. - Isolate, Substitute, Solve, Verify.

🎯 Super Acronyms

S.O.S. - Substitute, Obtain, Solve.

Examples

  • Example 1: Solve 7x - 15y = 2 and x + 2y = 3 using substitution.

  • Example 2: Aftab and his daughter's age problem utilizing substitution method to establish linear equations for their ages.

Glossary of Terms

  • Term: Substitution Method

    Definition:

    An algebraic method of solving linear equations by substituting a variable's value from one equation into the other.

  • Term: Linear Equation

    Definition:

    An equation that graphs as a straight line and can be expressed in the form ax + by = c.

  • Term: Isolate

    Definition:

    To express one variable in terms of others, typically to prepare for substitution.