📐 Linear Equations

What is a Linear Equation in One Variable? A linear equation in one variable is an equation that can be written in the form \(ax + b = 0\), where \(a\) and \(b\) are constants, \(a ≠ 0\), and \(x\) is the variable. The highest power of the variable is 1.

Examples:

• \(2x + 3 = 7\) (linear)
• \(5x - 4 = 6x + 2\) (linear)
• \(x² + 3 = 5\) (not linear – power is 2)

What does it mean to "solve" an equation? Solving an equation means finding the value(s) of the variable that make the equation true. This value is called the solution or root of the equation.

Rules for Solving Linear Equations:

RuleExplanationExampleAddition/Subtraction PropertyAdd or subtract same number from both sides\(x - 5 = 10 ⇒ x = 15\)Multiplication/Division PropertyMultiply or divide both sides by same non-zero number\(3x = 12 ⇒ x = 4\)Combining Like TermsSimplify each side before solving\(2x +…

What are Fractional Equations? Fractional equations contain fractions with the variable in the numerator, denominator, or both. To solve them, we eliminate denominators by multiplying both sides by the Least Common Multiple (LCM) of all denominators.

Steps to Solve Fractional Equations:

• Find the LCM of all denominators
• Multiply both sides of the equation by the LCM
• Simplify (cancel denominators)
• Solve the resulting linear equation
• Check that the solution doesn't make any denominator zero

How to Approach Word Problems:

StepAction1Read the problem carefully2Identify what is being asked (the unknown)3Choose a variable to represent the unknown4Translate the words into a mathematical equation5Solve the equation6Answer the question (with appropriate units)
Common Word Problem Types:

• Age problems: Compare ages now and in the future
• Number problems: Find…

What are Simultaneous Equations? Simultaneous equations are two or more equations that share the same variables. They are solved together (simultaneously) to find values that satisfy all equations at the same time.

Example: \(x + y = 10\) \(x - y = 4\)

Why do we need them? Sometimes one equation with two variables has infinitely many solutions. Adding another equation gives a unique solution.

Methods to Solve Simultaneous Equations:

MethodHow it worksBest forElimination MethodAdd/subtract equations to eliminate one variableWhen coefficients are friendlySubstitution MethodSolve one equation for one variable, substitute into otherWhen one variable is easily isolated
Elimination Method Steps:

• Align equations so like terms are stacked
• Multiply one or both equations to make coefficients of one variable opposites
• Add or subtract to eliminate that variable
•…