Chapter 3 linear systems
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  • 1. Chapter 3 Linear Systems
    3.1 Solving Systems Using Tables and Graphs
    3.2 Solving Systems Algebraically
  • 2. System of Equations
    A system of equations is a set of two or more equations
    A linear system consists of linear equations
    A solution of a system is a set of values for the variables that makes all the equations true.
    (usually an ordered pair)
    Systems can be solved be various methods: graphing, substitution, and elimination
  • 3. Solving a system by graphing
    Write each equation in slope-intercept form
    Graph each line
    Find the point of intersection (this is your solution)
    Check by substituting the values into both equations
  • 4. Solve each system by graphing
  • 5. Classifying Systems
    A system of two linear equations can be classified by the number of solutions it has
    A consistent systems has at least one solution
    An independent system has one solution
    A dependent system has infinitely many solutions
    An inconsistent system has no solution
  • 6. Without graphing, classify each system as independent, dependent, or inconsistent
    Rewrite each equation into slope-intercept form
    Compare the slopes and y-intercepts
    Different slopes: independent system
    Same slope and same y-intercept: dependent system
    Same slope and different y-intercept: inconsistent
  • 7. Without graphing, classify each system as independent, dependent, or inconsistent
  • 8. Without graphing, classify each system as independent, dependent, or inconsistent
  • 9. Solving Systems by Substitution
    Solve one equation for one of the variables
    Substitute the expression into the other equation and solve
    Substitute the solution into one of the original equations and solve for the remaining variable
    Check the solution
  • 10. Solving Systems by Substitution
    Use when it is easy to isolate one of the variables
  • 11. Solving by Elimination
    Rewrite both equations in standard form
    Multiply one or both systems by an appropriate non-zero number (note you want one variable to drop out in the next step)
    Add the equations
    Solve for the variable
    Substitute the value into one of the original equation and solve for the remaining variable
    Check the solution
  • 12. Solve by elimination
  • 13. Solving systems without unique Solutions
    Solving a system algebraically can sometimes lead to infinitely many solutions and/or no solution
    If you get a true result: infinitely many solutions
    If you get a false result: no solution
  • 14. Example: Solve the system
  • 15. Example: Solve the system