Defining Linear Equations

Defining Linear Equations

• A linear equation is an equation that can be plotted as a straight line on a graph.
• This means the variables involved are raised to the power of one (first degree).
• General form: Ax + By = C, where A, B, and C are constants, and x and y are variables.

Key Components of a Linear Equation

• Variables: Typically represented by letters like x and y. These represent unknown values.
• Constants: Numerical values (e.g., 2, -5, 10) that don't change.
• Coefficients: Numbers that multiply the variables (e.g., 3 in 3x).
• Slope (m): Represents the rate of change of y with respect to x. It dictates the steepness and direction of the line.
• Y-intercept (b): The point where the line crosses the y-axis. This is the value of y when x = 0.

Slope-Intercept Form

• The most common way to express a linear equation is in slope-intercept form: y = mx + b.
• m represents the slope
• b represents the y-intercept.

Standard Form

• Another way to express a linear equation is in standard form: Ax + By = C.
• A, B, and C are integers. A is usually positive.

Finding the Slope

• The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁).
• Important to note the order of subtraction in the formula.

Graphing Linear Equations

• To graph a linear equation, two points are typically needed to define the line.
• One common method is to find the x and y intercepts.
• Another method is to use the slope and y-intercept.

Parallel and Perpendicular Lines

• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals of each other.

Systems of Linear Equations

• A system of linear equations consists of two or more linear equations.
• Solving a system involves finding the values of the variables that satisfy all the equations simultaneously.
• This can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).

Solving Linear Equations

• Substitution Method: Replacing one variable in one equation with its expression from another equation.
• Elimination Method: Adding or subtracting equations to eliminate a variable.
• Graphing Method: Visually determining the point where the lines intersect.

Applications of Linear Equations

• Real-world problems involving constant rates of change can often be modeled with linear equations.
• Examples include calculating distance, cost, and profit.

This quiz covers the fundamental concepts of linear equations, including their definition, key components, and different forms such as slope-intercept and standard form. Test your knowledge on variables, constants, coefficients, slope, and y-intercept.