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Solving Systems of Equations by Elimination Warm Up 4/3/17 Solve each system by graphing. 1. 𝑦 = 1 𝑥 2 + 1.5 𝑦 = −2𝑥 + 4 (1, 2) 2. 3𝑥 + 𝑦 = −5 𝑦 = −5 (0, -5) Elimination To get rid of. When 2 variables have the same coefficients, they can be eliminated by subtraction. (ex. 2x – 2x = 0 When 2 variables have the same inverse coefficients, they can be eliminated by addition. (ex. 3y + -3y = 0) Independent Discovery Randy has some $1 bills and $5 bills in his wallet. He has 15 bills in all. He counts the money and finds he has $47. How many of each type of bill does Randy have? Write a system of equations for this problem. Step 1: Define the variables x = number of $1 bills y = number of $5 bills Step 2: Write the equations 1st equations represents the total amount of bills: 𝑥 + 𝑦 = 15 2nd equation represents the amount of money: 1𝑥 + 5𝑦 = 47 Solving by Elimination 1. Add the equations by adding the like terms. Describe the result. Can you solve the resulting equation? Why or why not? 𝑥 + 𝑦 = 15 𝑥 + 5𝑦 = 47 2. Subtract the equations by subtracting the like terms. Describe the result. Can you solve the resulting equation? Why or why not? Solving by Elimination (add/subtract) Step 1: Write each equation in Standard Form Ax + By = C Step 2: Add or subtract the equations. Remember to distribute the subtraction symbol to all terms. Step 3: Substitute the value of the variable you solved for back into an original equation. Solve for the other variable. Step 4: Check by substituting both answers into both equations. Solving a System by Adding Step 1: Eliminate one variable. Since the coefficients of y are additive inverses, add to eliminate y. 5x – 6y = -32 + 3x + 6y = 48 8x + 0 = 16 x=2 5x – 6y = -32 3x + 6y = 48 Solution: (2,7) Add Solve for x (Divide both sides by 8) Step 2: Substitute 2 for x in either equation and solve for y. 5x – 6y = -32 Write either equation 5(2) – 6y = -32 Substitute 2 for x 10 – 6y = -32 Subtract 10 from both sides -6y = -42 Simplify y = 7 Solve for y (Divide both sides by -6) Solving a System by Subtracting At the school store, Ricardo bought 4 pencils and 6 erasers and spent $2.60. Annabelle bought 4 pencils and 10 erasers and spent $3.80. Solve the system of equations to determine the cost of 1 pencil and the cost of 1 eraser. Step 1: Write the system of equations. Let p = the cost of each pencil, and let e = the cost of each eraser. 4p + 6e = 2.60 amount Ricardo spent 4p + 10e = 3.80 amount Annabelle spent Step 2: Subtract the equations. Since the coefficients of p are the same, subtract to eliminate p. 4p + 6e = 2.60 - (4p + 10e = 3.80) The cost of one eraser is $0.30 0 - 4e = - 1.20 Subtract e = 0.30 Solve for e (Divide both sides by -4) Step 2: Substitute 0.30 for e in either equation and solve for p. 4p + 10e = 3.80 Write either equation 4p + 10(0.30) = 3.80 Substitute 0.30 for e The cost of one 4p + 3 = 3.80 Simplify pencil is $0.20 4p = 0.80 Subtract 3 from each side p = 0.20 Solve for y (Divide both sides by 4) Solving a System by Subtracting Example 2 Franklin and Marianne sell gourmet cakes for a fundraiser. Franklin sells 2 large cakes and 6 small cakes for $190. Marianne sells 2 large cakes and 3 small cakes for $130. Find the cost of each small cake. Step 1: Write the system of equations. Let l = the cost of each large cake, and let s = the cost of each small cake. 2l + 6s = 190 amount Franklin raised 2l + 3s = 130 amount Marianne raised Step 2: Subtract the equations. Since the coefficients of l are the same, subtract to eliminate l. 2l + 6s = 190 - (2l + 3s = 130) 0 + 3s = 60 Subtract s = 20 Solve for e (Divide both sides by 3) The cost of one small cake is $20 Examples 1 and 2 show that in order to eliminate a variable, the coefficients of the variable must be the same or the additive inverse. Sometimes you have to multiply each side of one or both of the equations in a system by a nonzero number before you can eliminate a variable. (We will look at those examples tomorrow) CW/HW – Solve each system using elimination. Check your solution. 1. 𝑥 + 𝑦 = 9 𝑥−𝑦 =1 3. −𝑥 + 𝑦 = −2 3𝑥 − 𝑦 = 4 2. 3𝑥 + 2𝑦 = 2 𝑥 − 2𝑦 = 6 4. 3𝑥 + 𝑦 = 9 3𝑥 + 3𝑦 = 21 Solving Systems of Equations using Elimination Day 2 Warm Up 4/4/17 Solve each system by graphing. 1. 2𝑥 − 𝑦 = 1 𝑥 =𝑦+1 (0, -1) 2. 8𝑥 − 4𝑦 + 8 = 0 6x + 3𝑦 + 6 = 0 (-1, 0) Solving a Linear Equation by Multiplying First: Step 1: Write each equation in Standard Form (Ax + By = C) Step 2: Multiply/ divide one or both equations so one variable has identical or inverse coefficients Step 3: Add or subtract the equations. Remember to distribute the subtraction symbol to all terms. Step 4: Substitute the value of the variable you solved for back into an original equation. Solve for the other variable. Step 5: Check by substituting both answers into both equations. Solving a System by Multiplying Step 1: To eliminate x, multiply each term of the second equation by 2. Then add. 10x – 7y = 2 2(-5x + 3y = -3) = 2(-3) 10x – 7y = 2 -5x + 3y = -3 10x – 7y = 2 + (-10x + 6y = -6) 0 – 1y = -4 Add y = 4 Solve for y (Divide both sides by -1) Step 2: Substitute 4 for y in either original equation and solve for x. 10x – 7y = 2 Write either equation 10x – 7(4) = 2 Substitute 4 for y Solution: (3,4) 10x – 28 = 2 Simplify 10x = 30 Add 28 to both sides x = 3 Solve for x (Divide both sides by 10) Solve each system of equations by elimination. Show your work. 1. 2x + 6y = -22 4x – 3y = -14 (2, ½) 2. x – 4y = 2 3x + 5y = 40 (10, 2) 3. 4x – 2y = 7 3x + 6y = 9 (2, ½) 1. Which of the following systems would be most efficiently solved using the elimination method? System A: 2x + 2y = 6 -6x – 2y = 6 System B: y = 2 – 3x 4x – 2y = -2 System A – the coefficients of y are additive inverses of each other. 2. Explain how you would solve the following system using the elimination method. 2x – 5y = -6 2x – 7y = 14 Subtract the second equation from the first to eliminate x. Solve each system of equations by elimination. Show your work. 1. 2x + y = 12 x–y=3 (5, 2) 2. How do you know when to add the equations to eliminate a variable? When the coefficients of the same variable are positive and negative Exit Ticket On a separate sheet of paper to turn in. 1. Solve the system by elimination and check your solution. 5𝑥 − 6𝑦 = 1 2𝑥 + 2𝑦 = 18 2. Would you add or subtract the equations to solve the following system? Explain your reasoning. 5𝑥 + 3𝑦 = 4 5𝑥 − 3𝑦 = −16