Understanding Linear Equations

Understanding Linear Equations

• Linear equations represent a relationship between two variables where the graph of the relationship is a straight line.
• The general form of a linear equation is ax + by = c, where 'a', 'b', and 'c' are constants, and 'x' and 'y' are variables.
• A linear equation can also be written in slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept.
• The slope of a line represents the rate of change of 'y' with respect to 'x'. A positive slope indicates an upward trend, a negative slope a downward trend. A slope of zero indicates a horizontal line.
• The y-intercept is the point where the line crosses the y-axis. At this point, x = 0.
• Linear equations can be used to model a variety of real-world situations where the relationship between two variables is constant. For example, a car traveling at a constant speed.

Finding the Slope

• The slope can be calculated using two points (x₁, y₁) and (x₂, y₂) on a line.
• The formula for calculating the slope is: m = (y₂ - y₁) / (x₂ - x₁).
• Important considerations when using the slope formula:
  • If x₂ - x₁ = 0, the slope is undefined (a vertical line).
  • The order of the points matters; consistent subtraction is crucial (y₂-y₁)/(x₂-x₁).

Graphing Linear Equations

• Graphing a linear equation involves plotting points that satisfy the equation and connecting them to form a straight line.
• Using the slope-intercept form (y = mx + b), the y-intercept (b) provides a starting point for plotting.
• The slope (m) can be used to find additional points on the line. For example, if the slope is 2, move 2 units up and 1 unit to the right from the y-intercept to find another point.

Solving Linear Equations

• Solving a linear equation involves finding the value of the variable that makes the equation true.
• Basic algebraic manipulation is key.
• Different methods and approaches exist, but the goal is always to isolate the unknown variable.
• Examples include using addition, subtraction, multiplication, and division to isolate the variable.
• Checking the solution in the original equation is crucial for accuracy.

Systems of Linear Equations

• A system of linear equations consists of two or more linear equations with the same variables.
• Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system.
• Graphical solutions: The intersection point of the lines represents the solution.
• Algebraic solutions (substitution, elimination, graphing): These methods involve manipulating the equations to eliminate a variable or solve for a single variable.

Applications of Linear Equations

• Linear equations are used to model various real-world situations such as:
  • Calculating the total cost of items given a price per item and the quantity of items.
  • Finding the speed of an object knowing its distance travelled and the time taken.
  • Calculating the final temperature after a gradual heating or cooling process.
  • Analysing relationships between variables in various science concepts.
  • Determining budgets based on revenue and expenses.
  • Modelling the growth or decay of data.
• The applications of linear equations are broad and vary across many fields, including finance, engineering, physics, and business.