10 Questions
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 $f(x) = x^2 + 3x - 2$?
Quadratic function
What is the result of combining like terms in the algebraic expression $2x^2 + 3x + x^2 + 2x$?
$3x^2 + 5x$
What is the result of expanding the algebraic expression $(2x + 3)(x - 2)$?
$2x^2 + 5x - 6$
What is the type of algebraic expression that has the form $2x^2 + 3x - 1$?
Polynomial
What is the purpose of factorization in algebraic expressions?
To express an expression as a product of simpler expressions
What is a matrix?
A rectangular array of numbers
What is the advantage of using matrices to solve systems of linear equations?
It provides a compact and organized way to represent systems of linear equations
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).
Test your understanding of linear equations, including their general form, graphical representation, types, and methods for solving them.
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