Linear Equations

Definition and Notation

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• Linear equations can be written in the form: ax + by = c, where a, b, and c are constants, and x and y are variables.

Types of Linear Equations

• Simple Linear Equations: Equations with only one variable, e.g. 2x = 5.
• Linear Equations in Two Variables: Equations with two variables, e.g. 2x + 3y = 7.
• Linear Equations in Three Variables: Equations with three variables, e.g. x + 2y + 3z = 10.

Graphing Linear Equations

• The graph of a linear equation is a straight line.
• The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• The slope (m) represents the change in y over the change in x.
• The y-intercept (b) is the point where the line crosses the y-axis.

Solving Linear Equations

• Addition and Subtraction: Add or subtract the same value to both sides of the equation to isolate the variable.
• Multiplication and Division: Multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
• Graphical Method: Find the point of intersection of two lines on a graph to solve a system of linear equations.

Systems of Linear Equations

• A system of linear equations is a set of two or more linear equations with the same variables.
• Substitution Method: Solve one equation for one variable, then substitute it into the other equation to solve for the other variable.
• Elimination Method: Add or subtract equations to eliminate one variable, then solve for the other variable.

Definition and Notation of Linear Equations

• A linear equation has the highest power of the variable(s) as 1.
• Linear equations can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Types of Linear Equations

• Simple Linear Equations have only one variable, e.g. 2x = 5.
• Linear Equations in Two Variables have two variables, e.g. 2x + 3y = 7.
• Linear Equations in Three Variables have three variables, e.g.x + 2y + 3z = 10.

Graphing Linear Equations

• The graph of a linear equation is a straight line.
• The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• The slope (m) represents the change in y over the change in x.
• The y-intercept (b) is the point where the line crosses the y-axis.

Solving Linear Equations

• Addition and Subtraction Method: add or subtract the same value to both sides of the equation to isolate the variable.
• Multiplication and Division Method: multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
• Graphical Method: find the point of intersection of two lines on a graph to solve a system of linear equations.

Systems of Linear Equations

• A system of linear equations is a set of two or more linear equations with the same variables.
• Substitution Method: solve one equation for one variable, then substitute it into the other equation to solve for the other variable.
• Elimination Method: add or subtract equations to eliminate one variable, then solve for the other variable.