Learning Objectives
- Solve linear equations with variables on both sides
- Reduce equations to linear form
- Apply linear equations to solve word problems
- Understand the concept of transposition
Key Concepts
What is a Linear Equation?
A linear equation in one variable is an equation of the form ax + b = 0, where a and b are constants and a ≠ 0. The variable x has the power (degree) 1. Examples: 2x + 5 = 11, 3y - 7 = 8.
Solving Equations with Variables on Both Sides
When variables appear on both sides of the equation, bring all variable terms to one side and all constant terms to the other side.
Example: 5x - 3 = 3x + 7. Transpose 3x to the left: 5x - 3x = 7 + 3. Simplify: 2x = 10, so x = 5.
Transposition
Transposition means moving a term from one side of the equation to the other by changing its sign. When a positive term moves, it becomes negative, and vice versa. When a multiplying factor moves, it becomes a dividing factor.
Reducing Equations to Simpler Form
Some equations involve fractions. Multiply both sides by the LCM of the denominators to eliminate fractions, then solve normally.
Example: (x + 1)/2 = (2x - 1)/3. Multiply both sides by 6: 3(x + 1) = 2(2x - 1). Expand: 3x + 3 = 4x - 2. Solve: x = 5.
Cross-Multiplication Method
For equations of the form a/b = c/d, cross-multiply: a × d = b × c. This method simplifies equations involving fractions.
Applications (Word Problems)
Steps to solve word problems: (1) Read the problem carefully. (2) Identify the unknown and assign a variable. (3) Form an equation based on the given conditions. (4) Solve the equation. (5) Verify the answer.
Summary
Linear equations in one variable have the variable with degree 1. Equations can have variables on both sides, which are solved by transposition. Equations with fractions are simplified using LCM or cross-multiplication. Word problems are solved by forming and solving linear equations.
Important Terms
- Linear Equation: An equation where the highest power of the variable is 1
- Variable: A symbol (like x, y) representing an unknown value
- Transposition: Moving a term to the other side of an equation by changing its sign
- LCM: Least Common Multiple, used to eliminate fractions in equations
- Solution: The value of the variable that satisfies the equation
Quick Revision
- Linear equation: degree of variable is 1
- Transpose terms by changing their sign across the equals sign
- For fractional equations, multiply by LCM of denominators
- Cross-multiply when equation is in the form a/b = c/d
- Always verify the solution by substituting back into the original equation
- Word problems: identify unknown → form equation → solve → verify