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Lesson 2.8 Solving Systems of Equations by Elimination 1

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Key Concepts There are various methods to solving a system of equations. The three methods include the graphing method, the substitution method, and the elimination method. The elimination method involves eliminating one of the variables by using the properties of equality to multiply one or both of the equations by a constant and then add the equations. Solutions to systems are written as an ordered pair, (x,y). This is where the lines would cross if graphed. 2 2.2.1: Proving Equivalencies

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1.Multiply or divide one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables (sometimes this step can be skipped if your equations already differ only in sign for one of the variables). 2.Add the revised equations from Step 1, combining like terms to eliminate one of the variables. 3.Solve the resulting equation from Step 2. 4.Substitute the value obtained in Step 3 into either of the original equations and solve for the other variable. 3 2.2.1: Proving Equivalencies Steps for Elimination Method

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Guided Practice - Example 1 Solve the following system by elimination. 4 2.2.1: Proving Equivalencies 1.Multiply or divide one or both equations by a constant to obtain coefficients that differ only in sign for one of the variables. 3y and –3y are opposites, so step 1 can be skipped.

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5 2.2.1: Proving Equivalencies Simplify. Divide both sides by 3

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Guided Practice: Example 1, continued 4.Substitute the found value, x = 0, into either of the original equations to find the value of the other variable. 6 2.2.1: Proving Equivalencies 2x – 3y = –11First equation of the system 2(0) – 3y = –11Substitute 0 for x. – 3y = –11Simplify. Divide both sides by –3.

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Guided Practice: Example 1, continued 4.The solution to the system of equations is. If graphed, the lines would cross at. 7 2.2.1: Proving Equivalencies ✔

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Guided Practice - Example 2 Solve the following system by elimination. 8 2.2.1: Proving Equivalencies

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9 Divide both sides by -5 __________________

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10 2.2.1: Proving Equivalencies x + y = 1First equation of the system x + -2 = 1Substitute -2 for y. x = 3Add 2 to both sides.

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Guided Practice: Example 2, continued 4.The solution to the system of equations is (3, -2). If graphed, the lines would cross at (3, -2). 11 2.2.1: Proving Equivalencies ✔

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Guided Practice - Example 3 Solve the following system by elimination. 12 2.2.1: Proving Equivalencies 1.Multiply or divide one or both equations by a constant to obtain coefficients that differ only in sign for one of the variables. The variable x has a coefficient of 1 in the first equation and a coefficient of –2 in the second equation. Divide the second equation by 2. -2x + 6y = 4Original equation -x + 3y = 2Divide the equation by 2.

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13 2.2.1: Proving Equivalencies 0 + 0 = 7 0 = 7This is NOT a true statement. __________________

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Guided Practice: Example 3, continued The system does not have a solution. There are no points that will make both equations true. 14 2.2.1: Proving Equivalencies ✔

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