Integrated Algebra Notes: Solving systems by elimination Name Changing one equation Another algebraic method of solving a system is by elimination. Remember that equations need to be in one variable in order to be solved. Linear equations are often in two variables, x and y. In order to solve for one, we can eliminate the other. To use elimination, we set up one variable to have opposite coefficients. identify the variable we want to eliminate by looking at the coefficients identify the term we need to eliminate multiply one or both of the equations by the necessary factor combine the equations through addition To eliminate a variable, we need to decide which one would be the easiest to eliminate by looking at the coefficient of each. Looking at the equations: is either pair opposite coefficients already? Is either pair the same coefficient? Change one equation Is one a multiple of the other? Change one equation EX: Solve the system by elimination: No work needed x + 3y = 7 x - 4y = 14 The x coefficients are already the same number, 1. I need to change one equation to have an x coefficient of -1, then they will be opposites. I am going to leave the first equation alone, and multiply the second equation by a –1 to get my –1x. x + 3y = 7 x - 4y = 14 x + 3y = 7 -1 ( x - 4y = 14) Now, substitute -1 for y and solve for x. x + 3y = 7 x + 3(-1) = 7 x–3=7 x = 10 + x + 3y = 7 -x + 4y = -14 7y = -7 y = -1 solution: (10, -1) Integrated Algebra EX: Solve by elimination x + 2y = 7 3x - 2y = -3 What variable should we eliminate? (Look at coefficients) ____ Since the coefficients of the y are already opposites, I can get rid of them as they are. I do not need to change either equation. + x + 2y = 7 3x - 2y = -3 4x = 4 x=1 Substitute the x value in to find y. EX: Solve by elimination. 1 + 2y = 7 2y = 6 y=3 Solution: (1,3) 2x + 3y = 7 4x - 2y = -6 What variable should we eliminate? (Look at coefficients) Since the x’s have coefficients where one is a multiple of the other, I am going to eliminate the x’s. What coefficients do we need? ____ and _____ What equation do we change, first or second? _________ How do we change it? Multiply by _________ 2x + 3y = 7 4x - 2y = -6 -2(2x + 3y = 7) 4x – 2y = -6 Substitute 5 for y and find x: 4x – 2(5) = 6 4x – 10 = 6 4x = 16 x=4 -4x – 6y = -14 + 4x + 2y = - 6 -4y = -20 y = 5 Solution: (4,5) Integrated Algebra TRY: Solve by elimination 2x - 2y = 2 -3x + y = - 9 What variable should we eliminate? (Look at coefficients) ____ What coefficients do we need? ____ and _____ What equation do we change, first or second? _________ How do we change it? Multiply by _________ 2x - 2y = 2 -3x + y = -9 Substitute the x value in to find y. TRY: Solve by elimination 4x - 7y = -15 4x + 3y = 15 What variable should we eliminate? (Look at coefficients) ____ What coefficients do we need? ____ and _____ What equation do we change, first or second? _________ How do we change it? Multiply by _________ 4x - 7y = -15 4x + 3y = 15 Substitute the x value in to find y. Integrated Algebra HOMEWORK: Solving systems by elimination Changing one equation Solve each system by substitution. 1. x+y=3 x+y=3 2. 3x + y = 13 2x – y = 2 3. 2x – 3y = 5 3x + 6y = 4 4. x + 7y = 3 2x + y = 4 5. 3x – 2y = 10 2x – 2y = 5 6. 2x + 5y = 13 4x – 3y = -13 7. -5x + 8y = 21 10x + 3y = 15 8. 11x – 3y = 10 -2x + 3y = 8 Name

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