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Chapter 3: Systems of Equations A System of Equations is two or more equations using the same variables. - The solution to a system is the ordered pair (x, y) that satisfies BOTH of the equations. - There are 3 methods used to solve a system of equations 1. Graphing 2. Substitution 3. Elimination Solve by Graphing - Graph both lines to find the point of intersection. 1. One solution (x, y) 2. Parallel Lines have no solution 3. If both equations are graphed as the same line, there are infinitely many solutions Solve by Substitution -One of the equations in the system must be in the form x = something or y = something in order to substitute. EXAMPLE: Solve the system Substitute 2y in for x in the second equation and solve Now substitute y = 2 into the first equation to solve for x x = 2y 3y + 5x = 26 3y + 5(2y) = 26 3y + 10y = 26 13y = 26 so y=2 x = 2(2) so x=4 Therefore the solution point is (4, 2) Solving Systems by Elimination -If like terms are lined up in columns, and you see that one of the variables has opposite coefficients, you may add down the columns in order to ‘ELIMINATE’ that variable. EXAMPLE: Solve -The y terms eliminate, giving you an easy equation to solve for x. 2x + 3y = -7 + _________________ x – 3y = 19 3x = 12 So x=4 2(4) + 3y = -7 -Once you solve for x, substitute that value back into EITHER equation and solve for y. 8 + 3y = -7 3y = -15 So y = -5 Answer (4, -5) EXAMPLE 2: If no variables will eliminate, then you must multiply one or both of the equations by an integer that will create opposite coefficients for one of the variables. Solve -No term eliminated, therefore we had to multiply the first equation by -2 in order to cancel out the x terms -When the x terms eliminate, an easy equation is created to solve for y. x + 5y = 2 2x + 3y = -3 Multiply by -2 -2x – 10y = -4 + _________________ 2x + 3y = -3 -7y = -7 So y = 1 x + 5(1) = 2 -Once you solve for y, substitute that value into either of the original equations in order to solve for x. x+5=2 So x = -3 Answer (-3, 1) Section 3-3: Systems of Inequalities Review is a dashed boundary line , is a solid boundary line __________ < , > Shade the appropriate region above or below the boundary line. - In order to graph a system of inequalities, you must graph both inequalities and determine the OVERLAPPING SHADED REGION Example: Graph the system y x 1 y 2x 3 ** If there is no overlap, then there is no solution to the system.