Introduction to Linear Equations

Study Notes

Introduction to Linear Equations

A linear equation is an equation that can be written in the form ax + b = 0, where 'a' and 'b' are constants, and 'x' is a variable.
Linear equations represent a straight line on a graph.
Solving a linear equation involves finding the value of the variable that makes the equation true.

Key Concepts

Variable: A symbol (usually a letter) that represents an unknown numerical value.
Constant: A fixed numerical value.
Coefficient: The numerical factor that multiplies a variable.
Solution: The value of the variable that satisfies the equation.

Methods for Solving Linear Equations

Addition/Subtraction method: Add or subtract the same value to both sides of the equation to isolate the variable term.
Example: 2x + 5 = 11, subtract 5 from both sides to get: 2x = 6.
Multiplication/Division method: Multiply or divide both sides of the equation by the same value (not zero) to isolate the variable.
Example: (x/3) = 4, multiply both sides by 3 to get: x = 12.
Combining like terms: Simplify each side of the equation by combining like terms (variables with the same power).
Example: 3x + 2x - 5 = 10 becomes 5x - 5 = 10.

Solving Equations with Multiple Steps

Many equations require more than one step to solve.
Work through each step, keeping the equation balanced.
Follow the order of operations (PEMDAS/BODMAS), but in reverse when isolating the variable.
Example: 2(x-3) + 4 = 10
- Distribute: 2x - 6 + 4 = 10
- Simplify: 2x - 2 = 10
- Add 2 to both sides: 2x = 12
- Divide by 2: x = 6

Solving Equations with Fractions

Multiply both sides of the equation by the least common denominator (LCD) to eliminate fractions.
Example: (x/2) + 3 = 7
- Multiply by 2: x + 6 = 14
- Subtract 6: x = 8

Solving Equations with Parentheses

Distribute the number outside the parentheses to the terms inside.
Treat the expression inside the parenthesis as a single unit before isolating the variable.
Example: 2(x + 3) = 10
- Distribute: 2x + 6 = 10
- Subtract 6: 2x = 4
- Divide by 2: x = 2

Identifying Solutions

A solution to an equation is a value that makes the equation true.
Substitute the potential solution back into the original equation to verify. If the equation holds true, then your solution is correct.

Solving Equations with Variables on Both Sides

Move all the variable terms to one side of the equation and all the constant terms to the other side.
Example: 3x + 5 = x + 9
- Subtract x from both sides: 2x + 5 = 9
- Subtract 5 from both sides: 2x = 4
- Divide by 2: x = 2

Solving Equations with No Solutions or Infinitely Many Solutions

Some equations have no solution (inconsistent equations); they produce a false statement.
Some equations have infinitely many solutions (dependent equations); they produce an identity. Recognize these situations when simplifying.

Word Problems

Translate word problems into equations.
Define the variables.
Use the information given to create an equation.
Solve the equation.
Interpret your solution in the context of the word problem.

Example Problem

If 3 times a number, plus 7, is equal to 16, what is the number?
- Let x = the number
- Equation: 3x + 7 = 16
- Solving the equation: 3x = 9, x = 3.

Description

This quiz covers the fundamentals of linear equations, including their definition, key concepts, and methods for solving them. Explore the principles that govern solving equations of the form ax + b = 0, and learn various techniques such as addition, subtraction, multiplication, and division methods.