Linear Equations in Algebra

Definition and Terminology

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• It can be written in the form: ax + by = c, where a, b, and c are constants, and x and y are variables.
• The coefficients of the variables (a and b) can be any real numbers, including fractions and decimals.

Types of Linear Equations

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

Properties of Linear Equations

• Linearity: If x and y are solutions to a linear equation, then ax + by is also a solution, where a and b are constants.
• Homogeneity: If x is a solution to a linear equation, then kx is also a solution, where k is a constant.

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 standard form of a linear equation is Ax + By = C, where A, B, and C are integers.

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 linear equations to solve a system of linear equations.

Applications of Linear Equations

• Real-World Problems: Linear equations can be used to model real-world scenarios, such as calculating the cost of goods, the area of a rectangle, and the slope of a ramp.
• Science and Engineering: Linear equations are used to describe the motion of objects, the growth of populations, and the behavior of electrical circuits.

Linear Equations

• A linear equation is an equation in which the highest power of the variable(s) is 1 and can be written in the form: ax + by = c.
• The coefficients of the variables (a and b) can be any real numbers, including fractions and decimals.

Types of Linear Equations

• Simple linear equations have one variable, e.g., 2x = 5.
• Linear equations with two variables have two variables, e.g., 2x + 3y = 7.

Properties of Linear Equations

• Linearity: if x and y are solutions to a linear equation, then ax + by is also a solution, where a and b are constants.
• Homogeneity: if x is a solution to a linear equation, then kx is also a solution, where k is a constant.

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 standard form of a linear equation is Ax + By = C, where A, B, and C are integers.

Solving Linear Equations

• The addition and subtraction method involves adding or subtracting the same value to both sides of the equation to isolate the variable.
• The multiplication and division method involves multiplying or dividing both sides of the equation by the same non-zero value to isolate the variable.
• The graphical method involves finding the point of intersection of two linear equations to solve a system of linear equations.

Applications of Linear Equations

• Linear equations can be used to model real-world scenarios, such as calculating the cost of goods, the area of a rectangle, and the slope of a ramp.
• Linear equations are used in science and engineering to describe the motion of objects, the growth of populations, and the behavior of electrical circuits.