Linear Equations in Algebra

Study Notes

A linear equation is an equation in which the highest power of the variable(s) is 1.
General form: ax + by = c, where a, b, and c are constants, and x and y are variables.
Linear equations can be represented graphically as straight lines.
Types of linear equations:
- Simple linear equations: e.g. 2x = 5
- Simultaneous linear equations: e.g. 2x + 3y = 7, x - 2y = -3
Methods for solving linear equations:
- Substitution method
- Elimination method
- Graphical method

A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
Notation: f(x) = ...
Domain: the set of input values for which the function is defined.
Range: the set of output values of the function.
Types of functions:
- Linear functions: e.g. f(x) = 2x + 1
- Quadratic functions: e.g. f(x) = x^2 + 3x - 2
- Exponential functions: e.g. f(x) = 2^x
Function operations:
- Composition: f(g(x))
- Inverse: f^(-1)(x)

An algebraic expression is a mathematical expression that can be simplified using algebraic operations.
Types of algebraic expressions:
- Monomials: e.g. 2x^2, 3y
- Binomials: e.g. 2x + 3, x^2 - 4
- Polynomials: e.g. 2x^2 + 3x - 1, x^3 - 2x^2 - 5x
Algebraic operations:
- Simplification: combining like terms
- Expansion: multiplying out brackets
- Factorization: expressing an expression as a product of simpler expressions

A matrix is a rectangular array of numbers, symbols, or expressions.
Notation: [a, b; c, d] or [[a, b], [c, d]]
Types of matrices:
- Square matrix: number of rows = number of columns
- Rectangular matrix: number of rows ≠ number of columns
- Identity matrix: a square matrix with all elements on the main diagonal = 1, and all other elements = 0
Matrix operations:
- Addition: element-wise addition of two matrices
- Multiplication: row-by-column multiplication of two matrices
- Inverse: a matrix that, when multiplied by another matrix, results in the identity matrix

Linear equations have the highest power of the variable(s) as 1.
The general form of a linear equation is ax + by = c, where a, b, and c are constants, and x and y are variables.
Graphically, linear equations are represented as straight lines.
There are two types of linear equations: simple linear equations (e.g. 2x = 5) and simultaneous linear equations (e.g. 2x + 3y = 7, x - 2y = -3).
Three methods for solving linear equations are: substitution method, elimination method, and graphical method.

A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
The notation for a function is f(x) =...
The domain of a function is the set of input values for which the function is defined.
The range of a function is the set of output values of the function.
There are three types of functions: linear functions (e.g. f(x) = 2x + 1), quadratic functions (e.g. f(x) = x^2 + 3x - 2), and exponential functions (e.g. f(x) = 2^x).
Two operations that can be performed on functions are: composition (f(g(x))) and inverse (f^(-1)(x)).

An algebraic expression is a mathematical expression that can be simplified using algebraic operations.
There are three types of algebraic expressions: monomials (e.g. 2x^2, 3y), binomials (e.g. 2x + 3, x^2 - 4), and polynomials (e.g. 2x^2 + 3x - 1, x^3 - 2x^2 - 5x).
Three algebraic operations that can be performed on expressions are: simplification (combining like terms), expansion (multiplying out brackets), and factorization (expressing an expression as a product of simpler expressions).

A matrix is a rectangular array of numbers, symbols, or expressions.
Matrices can be represented as [a, b; c, d] or [[a, b], [c, d]].
There are three types of matrices: square matrices (number of rows = number of columns), rectangular matrices (number of rows ≠ number of columns), and identity matrices (a square matrix with all elements on the main diagonal = 1, and all other elements = 0).
Three operations that can be performed on matrices are: addition (element-wise addition of two matrices), multiplication (row-by-column multiplication of two matrices), and inverse (a matrix that, when multiplied by another matrix, results in the identity matrix).