Defining Linear Equations

Defining Linear Equations

• A linear equation is an equation that can be graphed as a straight line.
• It represents a relationship between two variables, usually denoted as x and y.
• The general form of a linear equation is Ax + By = C, where A, B, and C are constants, and A and B are not both zero.

Key Components of Linear Equations

• Variables: Usually represented by 'x' and 'y' (or other letters), these represent unknown values.
• Constants: Numerical values that do not change in the equation. Examples include numbers like 2, -5, or 10.
• Coefficients: Numerical values that multiply the variables. For instance, in the equation 2x + 3y = 10, 2 and 3 are the coefficients of x and y, respectively.
• Slope: Represents the steepness of the line. It is calculated as the change in 'y' divided by the change in 'x' (rise over run).
• Y-intercept: The point where the line crosses the y-axis. This is the value of 'y' when 'x' is zero.
• X-intercept: The point where the line crosses the x-axis.

Standard Form and Slope-Intercept Form

• Standard Form: Ax + By = C. This form is useful for identifying the slope and intercepts.
• Slope-Intercept Form: y = mx + b. This form, where 'm' is the slope and 'b' is the y-intercept, is often the easiest to graph.

Finding the Slope

• The slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁)

Graphing Linear Equations

• Plotting points: Determine at least two points on the line by substituting values for 'x' and solving for 'y' (or vice versa). Plot these points on a coordinate plane and draw a straight line through them.
• Using the slope and y-intercept: If the equation is in slope-intercept form (y = mx + b), the y-intercept is the point (0, b). The slope (m) determines the steepness and direction of the line.

Solving Linear Equations

• Solving for 'x' or 'y': Isolate the variable you want to solve for on one side of the equation, often by performing inverse operations (addition, subtraction, multiplication, division) on both sides of the equation to eliminate constants and coefficients.
• Example Solve for 'x': 2x + 5 = 11
  • Subtract 5 from both sides: 2x = 6
  • Divide both sides by 2: x = 3

Applications of Linear Equations

• Representing relationships between quantities in various contexts (e.g., distance, time, cost).
• Modeling real-world situations that involve constant rates of change.
• Analyzing and predicting future values based on given information.
• Used extensively in business, economics, science, and many other areas.