## Questions and Answers

What is the general form of a linear equation in two variables?

$ax + by = c$

What is the method of solving a linear equation by replacing one variable with an expression in terms of the other variable?

Substitution method

What is the range of a function $f(x) = 2x + 1$?

All real numbers

What is the type of function that has the form $f(x) = x^2 + 3x - 2$?

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What is the result of combining like terms in the algebraic expression $2x^2 + 3x + x^2 + 2x$?

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What is the result of expanding the algebraic expression $(2x + 3)(x - 2)$?

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What is the type of algebraic expression that has the form $2x^2 + 3x - 1$?

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What is the purpose of factorization in algebraic expressions?

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What is a matrix?

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What is the advantage of using matrices to solve systems of linear equations?

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## Study Notes

### Linear Equations

- 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

### Functions

- 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)

### Algebraic Expressions

- 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

### Matrices

- 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

- 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.

### Functions

- 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)).

### Algebraic Expressions

- 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).

### Matrices

- 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).

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## Description

Test your understanding of linear equations, including their general form, graphical representation, types, and methods for solving them.