Linear Equations and Slope Basics

Defining Linear Equations

• A linear equation is an equation that can be graphically represented by a straight line.
• These equations typically involve variables to the first power (no exponents greater than 1).
• The standard form of a linear equation in two variables is Ax + By = C, where A, B, and C are constants, and x and y are variables.
• A more commonly used form is the slope-intercept form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Slope

• The slope of a line represents the rate of change between the y-values and the x-values as you move along the line.
• It is calculated as the change in 'y' (rise) divided by the change in 'x' (run): m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line.
• A positive slope indicates that the line rises from left to right.
• A negative slope indicates that the line falls from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope indicates a vertical line.

Y-intercept

• The y-intercept is the point where the line crosses the y-axis.
• In the equation y = mx + b, the y-intercept is represented by 'b'.
• To find the y-intercept, set x = 0 in the equation and solve for y.

Finding the Equation of a Line

• Given two points on the line:
  • First, calculate the slope using the formula above.
  • Then, substitute the slope and one of the points into the slope-intercept form (y = mx + b).
  • Solve for 'b' (the y-intercept).
  • Write the final equation using the calculated values for 'm' and 'b'.
• Given the slope and a point on the line:
  • Substitute the slope and the coordinates of the point into the slope-intercept form (y = mx + b).
  • Solve for 'b' (the y-intercept).
  • Write the final equation using the calculated values for 'm' and 'b'.
• Given the y-intercept and a point on the line:
  • Substitute the y-intercept into the slope-intercept form (y = mx + b).
  • Calculate the slope using the y-intercept and the given point.
  • Substitute the slope and the y-intercept into the slope-intercept form.

Graphing Linear Equations

• To graph a linear equation, you can:
  • Find the y-intercept and use the slope to find additional points.
  • Find two points on the line and draw a line through them.

Solving Linear Equations

• Solving a linear equation means finding the value of the variable that makes the equation true.
• To solve linear equations effectively, you must understand and apply the principles of equality. This means applying the same operations to both sides of the equation.
• Isolating the variable by performing appropriate addition, subtraction, multiplication, and division operations.

Applications of Linear Equations

• Linear equations are extensively used in various fields, such as:
  • Finance (budgeting, calculating interest)
  • Business (cost and revenue analysis)
  • Science (describing physical relationships)
  • Engineering (designing structures)
  • Everyday situations (calculating distance, speed, and time)
• Understanding linear equations empowers you to model and solve numerous real-world problems that involve relationships between two variables.