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# Linear Equations in Algebra

Created by
@BrilliantHappiness

## 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

All real numbers

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

<p>Quadratic function</p> Signup and view all the answers

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

<p>$3x^2 + 5x$</p> Signup and view all the answers

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

<p>$2x^2 + 5x - 6$</p> Signup and view all the answers

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

<p>Polynomial</p> Signup and view all the answers

### What is the purpose of factorization in algebraic expressions?

<p>To express an expression as a product of simpler expressions</p> Signup and view all the answers

### What is a matrix?

<p>A rectangular array of numbers</p> Signup and view all the answers

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

<p>It provides a compact and organized way to represent systems of linear equations</p> Signup and view all the answers

## 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.

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