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Chapter 3 Linear Systems 3.1 Solving Systems Using Tables and Graphs 3.2 Solving Systems Algebraically
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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
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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
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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
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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
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Without graphing, classify each system as independent, dependent, or inconsistent
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Without graphing, classify each system as independent, dependent, or inconsistent
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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
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Solving Systems by Substitution Use when it is easy to isolate one of the variables
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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
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Solve by elimination
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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