Solving Equations having the Variable on both Sides

2.2 Solving Equations having the Variable on both Sides

Description

Quick Overview

This section focuses on solving equations with variables on both sides, detailing methods and providing examples.

Standard

In this section, we learn how to solve equations where variables are present on both sides. We explore various methods to manipulate these equations, illustrated by examples that show the step-by-step process to isolate the variable and obtain solutions.

Detailed

Solving Equations with the Variable on Both Sides

An equation represents the equality of the values of two expressions, such as in the example 2x - 3 = 7. Here, the left-hand side (LHS) and the right-hand side (RHS) can also include variables, as demonstrated in 2x - 3 = x + 2.

To solve these equations, we need to isolate the variable. The section introduces how to manipulate both sides of the equation. For example, in the equation 2x - 3 = x + 2, steps such as subtracting x from both sides help us to eventually isolate x as shown in the solution where it simplifies to x = 5.

Additional examples, such as 5x + 7/2 = x - 14, illustrate multiplying each side of the equation to simplify the expressions further. The section emphasizes understanding the operations performed on both sides to maintain equality and reach the solution effectively.

Example 2: Solve \[ 3x - 4 = 2x + 5 \]

Solution: We have

\[ 3x - 4 = 2x + 5 \]

or

\[ 3x - 2x = 5 + 4 \]

or

\[ x = 9 \]

(solution)

Key Concepts

  • Isolating Variables: The process of getting the variable on one side of the equation.

  • Maintaining Equality: Understanding that operations must be balanced on both sides.

  • Step-by-Step Solutions: Breaking down each operation to find the solution.

Memory Aids

🎵 Rhymes Time

  • To keep the balance, let's not lose, do the same to both, that's the right move!

📖 Fascinating Stories

  • Once there was a magician named Variable who loved to play games. He would always keep EQUALITY balanced by doing the same trick on both sides of his magic equation.

🧠 Other Memory Gems

  • Remember: I.S.O.L.A.T.E - Isolate, Subtract, Or, Leave Alone The Equation!

🎯 Super Acronyms

MATH - Move All Terms Homogeneously.

Examples

  • Example 1: Solve 2x - 3 = x + 2, leading to x = 5.

  • Example 2: Solve 5x + 7/2 = x - 14, leading to x = -35/8.

Glossary of Terms

  • Term: Equation

    Definition:

    A statement that asserts the equality of two expressions, typically containing variables.

  • Term: Variable

    Definition:

    A symbol representing an unknown quantity in mathematics, often denoted by letters such as x, y, etc.