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Algebra 2 – Unit 13 – Systems of Equations - NEED TO KNOW….. A. Terminology – a. System of Linear Equations is 2 or more equations considered together. b. Solution of a System of Linear Equations is the ordered pair ( x, y ) (if it exists) that when substituted into each linear equation in the system of linear equations makes each TRUE. c. Independent System of Linear Equations – When a system of equations has only one solution it is called Independent. The graph of the system will show equations intersecting at only one point. d. Inconsistent System of Linear Equations – When a system of equations has no intersecting points then there is no solution and it is called Inconsistent. The graph of the system will show parallel lines. e. Dependent System of Linear Equations – When all solutions for one equation work for all the other equations in the system the system is called Dependent. The graph of the system will be the same lines. The solution looks like the ordered pair, ( x, y solved in terms of x) . f. Matrix – A rectangular array of numbers. Each number in the matrix is called an element. Each element is denoted by its row and column. a11 a12 a This is called a 2 x 2 matrix or a square matrix. 21 a22 g. Determinant – A number that is associated with a matrix. It is a a a formula: 11 12 (a11 )(a22 ) (a21 )(a12 ) a21 a22 h. Cramer’s Rule – Method to solve a System of Linear Equations. If you have: a11 x a12 y b1 a21 x a22 y b2 Then the solution (x, y ) is found by ( x D= a11 a12 a21 a22 , Dx b1 a12 b2 a22 , Dy a11 b1 a21 b2 Dy Dx ,y ) where D D , & D 0. i. Systems of Linear Inequalities are solved Graphically by shading in the region of intersection if one exists once the inequalities are graphed. B. Methods to Solve Systems of Linear Equations – a. Graphing Method - Just graph the equations and find the point of intersection, if one exists. The solution is the point of intersection, an ordered pair ( x, y ) . b. Substitution Method – Solve one of the equations in the system for one of the variables. Then substitute into the other equation for that variable and solve. Take that variable solution and solve to find the value of the other variable. c. Addition Method – The goal using this method is to get rid of one of the variables by adding the equations together. To do this you may have to multiply one or both of the equations by something so when you add the equations together one of the variables will cancel out. Once a variable is cancelled out then use Substitution to finish the problem. d. Cramer’s Rule - Method to solve a System of Linear Equations. If you have a system of linear equations that looks like: a11 x a12 y b1 a21 x a22 y b2 Then the solution (x, y ) is found by ( x D= a11 a12 a21 a22 , Dx b1 a12 b2 a22 , Dy a11 b1 a21 b2 Dy Dx ,y ) where D D , & D 0. C. Method to Solve Systems of Inequalities - Systems of Linear Inequalities are solved Graphically by shading in the region of intersection, if one exists, once the inequalities are graphed. The following websites demonstrate how to graph and solve systems of inequalities: http://www.purplemath.com/modules/syslneq.htm - This website gives good examples. http://www.mathwarehouse.com/algebra/linear_equation/systems-ofequation/system-linear-inequality.php - This website is interactive.