Solving Systems of Equations by Elimination

Warm Up 4/3/17 (1, 2) (0, -5) Solve each system by graphing. 𝑦= 1 2 𝑥+1.5 3𝑥+𝑦=−5 𝑦=−5 𝑦=−2𝑥+4 (1, 2) (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 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 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 5x – 6y = -32 3x + 6y = 48 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  Add x = 2  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) Solution: (2,7)

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

The cost of one eraser is $0.30 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) 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 4p + 3 = 3.80  Simplify 4p = 0.80  Subtract 3 from each side p = 0.20  Solve for y (Divide both sides by 4) The cost of one eraser is $0.30 The cost of one pencil is $0.20

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

The cost of one small cake is $20 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) 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) The cost of one small cake is $20

CW/HW – Solve each system using elimination. Check your solution. 1. 𝑥+𝑦=9 𝑥−𝑦=1 2. 3𝑥+2𝑦=2 𝑥−2𝑦=6 3. −𝑥+𝑦=−2 3𝑥 −𝑦=4 4. 3𝑥+ 𝑦=9 3𝑥+3𝑦=21 1. (5, 4) 2. (2, -2) 3. (1, -1) 4. (1, 6)

Solving Systems of Equations using Elimination Day 2

Warm Up 4/4/17 (0, -1) (-1, 0) Solve each system by graphing. 2𝑥 −𝑦=1 8𝑥−4𝑦+8=0 𝑥=𝑦+1 6x+3𝑦+6=0 (0, -1) (-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 10x – 7y = 2 -5x + 3y = -3 Step 1: To eliminate x, multiply each term of the second equation by 2. Then add. 10x – 7y = 2  10x – 7y = 2 2(-5x + 3y = -3) = 2(-3)  + (-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 10x – 28 = 2 Simplify 10x = 30 Add 28 to both sides x = 3  Solve for x (Divide both sides by 10) Solution: (3,4)

Solve each system of equations by elimination. Show your work. 1. 2x + 6y = -22 4x – 3y = -14 2. x – 4y = 2 3x + 5y = 40 3. 4x – 2y = 7 3x + 6y = 9 (2, ½) (10, 2)

System A: 2x + 2y = 6 System B: y = 2 – 3x -6x – 2y = 6 4x – 2y = -2 1. Which of the following systems would be most efficiently solved using the elimination method? System A: 2x + 2y = 6 System B: y = 2 – 3x -6x – 2y = 6 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.

When the coefficients of the same variable are positive and negative Solve each system of equations by elimination. Show your work. 1. 2x + y = 12 x – y = 3 2. How do you know when to add the equations to eliminate a variable? (5, 2) 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 (-1, 3)