Linear Equations & Graphs

Linear Equations & Graphs

Linear Equations:

  • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
  • Linear equations can have one, two, three or more variables. When it only involves one variable, it is termed a simple linear equation. For instance, 2x + 3 = 0.
  • When it involves two variables, it is illustrated as an equation of a straight line when graphed. The standard form: Ax + By = C, where A, B, and C are constants.
  • Slope-intercept form is another common form of linear equations, expressed as y=mx+c, where m is the slope and c is the y-intercept.

Features of Linear Graphs:

  • A line graph portrays information as a series of data points connected by straight line segments.
  • Every linear equation produces a straight line when graphed.
  • You can identify a linear graph by its straight line shape.
  • Slope indicates the steepness of the line, it shows the change in y for each unit increase in x.
  • The y-intercept is the point where the line crosses the y-axis.
  • x-intercept is where the line crosses the x-axis.
  • The slope and y-intercept allow us to quickly interpret and sketch graphs.

Techniques for Sketching Linear Graphs:

  • Graph linear equations by plotting points. Choose any values for x and then calculate the corresponding y-values.
  • Use the y = mx + c form to quickly determine the slope and y-intercept. Begin at the intercept (when x = 0) and use the slope to find other points.
  • Draw a straight line through your points.

Solving Linear Equations:

  • Solving a linear equation usually means finding the value of y for a given x.
  • To solve for y in terms of x (in the form y = mx + c), rearrange the equation.
  • Solving systems of linear equations can be accomplished by graphing the equations and finding the intersection points.
  • Other methods for solving systems of linear equations include substitution method and elimination method.

Applications of Linear Equations:

  • Linear equations can be used for calculating distances, converting temperatures, and estimating sales, among other real-world scenarios.
  • They help in finding relationships between two variables, predicting results and making predictions.
  • Linear equations are widespread in science, business, architecture, and other fields.

Remember, practice makes perfect! Keep working on equations and drawing graphs to improve your understanding and skills.