"A linear equation is an equation that may be put in the form... where x1, ..., xn are the variables (or unknowns), and b, a1, ..., an are the coefficients..." Quote: "a linear equation is an equation that may be put in the form a1x1 + ... + anxn + b = 0"
Solving linear equations involves finding the value of an unknown variable in an equation with one variable.
Variables: The concept of variables and how they are used in algebraic expressions.
Solving Linear Equations: The basic steps to solve a simple linear equation.
Basic Operations: Addition, subtraction, multiplication and division of algebraic expressions.
Distributive Property: The distributive property of multiplication over addition.
Combining Like Terms: How to combine like terms in an equation to simplify it.
Linear Inequalities: Understanding and solving linear inequalities.
Absolute Value Equations: Solving absolute value equations and inequalities.
Systems of Equations: Solving systems of linear equations by substitution, elimination or graphing.
Graphing Linear Equations: Plotting the graph of a linear equation.
Word Problems: Understanding and solving word problems involving linear equations.
Rational Expressions: Simplifying and solving equations with rational expressions.
Quadratic Equations: An introduction to solving quadratic equations.
One-step equations: Simple equations that can be solved in one step by applying inverse operations (adding or subtracting a number, multiplying or dividing by a number).
Two-step equations: Equations that require two steps to solve, which typically involves applying opposite operations in the order of PEMDAS (parentheses, exponents, multiplication/division, addition/subtraction).
Multi-step equations: Equations that require three or more steps to solve, which are typically solved by combining like terms, simplifying expressions, and using inverse operations to isolate the variable.
Equations with fractions: Equations that include rational expressions that require multiplying both sides of the equation by the lowest common denominator to eliminate the fractions and solve for the variable.
Equations with decimals: Equations that include decimal numbers that can be solved in the same way as equations with whole numbers, by isolating the variable and solving for its value.
Equations with absolute values: Equations that contain an absolute value expression, which requires considering both possible values (positive and negative) of the absolute value expression separately and setting up two equations to solve for the variable.
Equations with variables on both sides: Equations in which the variable term appears on both sides of the equation, requiring the use of inverse operations to simplify the equation and isolate the variable on one side.
Equations with variables in denominators: Equations that include variable expressions in denominators, which require multiplying both sides of the equation by the common denominator to eliminate the denominators and solve for the variable.
Equations with variables in exponents: Equations that include variable expressions raised to powers, which require taking logarithms or roots of both sides of the equation to isolate the variable and solve for its value.
Systems of linear equations: Equations that include two or more variables and require solving for the values of all variables that satisfy both equations simultaneously, typically using substitution, elimination, or graphing methods.
"The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables." Quote: "The coefficients may be considered as parameters of the equation..."
"The coefficients are required to not all be zero." Quote: "...the coefficients are required to not all be zero."
"The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true." Quote: "The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true."
"In the case of just one variable, there is exactly one solution (provided that a1 ≠ 0)." Quote: "In the case of just one variable, there is exactly one solution..."
"The solutions of a linear equation form a line in the Euclidean plane..." Quote: "The solutions of a linear equation form a line in the Euclidean plane..."
"Conversely, every line can be viewed as the set of all solutions of a linear equation in two variables." Quote: "Conversely, every line can be viewed as the set of all solutions of a linear equation in two variables."
"The solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n." Quote: "The solutions of a linear equation in n variables form a hyperplane..."
"Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations." Quote: "Linear equations occur frequently in all mathematics and their applications in physics and engineering..."
"This article considers the case of a single equation with coefficients from the field of real numbers, for which one studies the real solutions." Quote: "This article considers the case of a single equation with coefficients from the field of real numbers..."
"All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field." Quote: "All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field."
"The term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown." Quote: "Often, the term linear equation refers implicitly to this particular case..."
"The solutions of a linear equation in n variables form a hyperplane..." Quote: "The solutions of a linear equation in n variables form a hyperplane..."
"Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations." Quote: "Linear equations occur frequently in all mathematics and their applications in physics and engineering..."
"For the case of several simultaneous linear equations, see system of linear equations." Quote: "For the case of several simultaneous linear equations, see system of linear equations."
"The coefficients may be considered as parameters of the equation..." Quote: "The coefficients may be considered as parameters of the equation..."
"x1, ..., xn" Quote: "...where x1, ..., xn are the variables (or unknowns)..."
"The coefficients may be arbitrary expressions, provided they do not contain any of the variables." Quote: "...provided they do not contain any of the variables."
"The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true." Quote: "The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true."
"In the case of just one variable, there is exactly one solution (provided that a1 ≠ 0)." Quote: "In the case of just one variable, there is exactly one solution..."