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Objective The student will be able to: solve systems of equations using elimination with addition and subtraction.

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Solving Systems of Equations So far, we have solved systems using graphing and substitution. These notes show how to solve the system algebraically using ELIMINATION with addition and subtraction. Elimination is easiest when the equations are in standard form. Standard Form: ax + by = c

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Solving a system of equations by elimination using addition and subtraction. Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Add or subtract the equations. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.

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1) Solve the system using elimination. x + y = 5 3x – y = 7 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! The y’s have the same coefficient. Step 3: Add or subtract the equations. Add to eliminate y. x + y = 5 (+) 3x – y = 7 4x = 12 x = 3

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1) Solve the system using elimination. Step 4: Plug back in to find the other variable. x + y = 5 (3) + y = 5 y = 2 Step 5: Check your solution. (3, 2) (3) + (2) = 5 3(3) - (2) = 7 The solution is (3, 2). What do you think the answer would be if you solved using substitution? x + y = 5 3x – y = 7

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2) Solve the system using elimination. 4x + y = 7 4x – 2y = -2 Step 1: Put the equations in Standard Form. They already are! Step 2: Determine which variable to eliminate. The x’s have the same coefficient. Step 3: Add or subtract the equations. Subtract to eliminate x. 4x + y = 7 (-) 4x – 2y = -2 3y = 9 y = 3 Remember to “keep-change- change”

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2) Solve the system using elimination. Step 4: Plug back in to find the other variable. 4x + y = 7 4x + (3) = 7 4x = 4 x = 1 Step 5: Check your solution. (1, 3) 4(1) + (3) = 7 4(1) - 2(3) = -2 4x + y = 7 4x – 2y = -2

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Which step would eliminate a variable? 3x + y = 4 3x + 4y = 6 1. Isolate y in the first equation 2. Add the equations 3. Subtract the equations 4. Multiply the first equation by -4

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Solve using elimination. 2x – 3y = -2 x + 3y = 17 1. (2, 2) 2. (9, 3) 3. (4, 5) 4. (5, 4)

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3) Solve the system using elimination. y = 7 – 2x 4x + y = 5 Step 1: Put the equations in Standard Form. 2x + y = 7 4x + y = 5 Step 2: Determine which variable to eliminate. The y’s have the same coefficient. Step 3: Add or subtract the equations. Subtract to eliminate y. 2x + y = 7 (-) 4x + y = 5 -2x = 2 x = -1

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2) Solve the system using elimination. Step 4: Plug back in to find the other variable. y = 7 – 2x y = 7 – 2(-1) y = 9 Step 5: Check your solution. (-1, 9) (9) = 7 – 2(-1) 4(-1) + (9) = 5 y = 7 – 2x 4x + y = 5

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What is the first step when solving with elimination? 1. Add or subtract the equations. 2. Plug numbers into the equation. 3. Solve for a variable. 4. Check your answer. 5. Determine which variable to eliminate. 6. Put the equations in standard form.

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Find two numbers whose sum is 18 and whose difference 22. 1. 14 and 4 2. 20 and -2 3. 24 and -6 4. 30 and 8

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Day 2

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Solving Systems of Equations So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication. What happens when the coefficients are not the same? We multiply the equations to make them the same! You’ll see…

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Solving a system of equations by elimination using multiplication. Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Multiply the equations and solve. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.

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1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! None of the coefficients are the same! Find the least common multiple of each variable. LCM = 6x, LCM = 2y Which is easier to obtain? 2y (you only have to multiply the bottom equation by 2)

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1) Solve the system using elimination. Step 4: Plug back in to find the other variable. 2(2) + 2y = 6 4 + 2y = 6 2y = 2 y = 1 2x + 2y = 6 3x – y = 5 Step 3: Multiply the equations and solve. Multiply the bottom equation by 2 2x + 2y = 6 (2)(3x – y = 5) 8x = 16 x = 2 2x + 2y = 6 (+) 6x – 2y = 10

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1) Solve the system using elimination. Step 5: Check your solution. (2, 1) 2(2) + 2(1) = 6 3(2) - (1) = 5 2x + 2y = 6 3x – y = 5 Solving with multiplication adds one more step to the elimination process.

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2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 Step 1: Put the equations in Standard Form. They already are! Step 2: Determine which variable to eliminate. Find the least common multiple of each variable. LCM = 4x, LCM = 12y Which is easier to obtain? 4x (you only have to multiply the top equation by -4 to make them inverses)

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2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 Step 4: Plug back in to find the other variable. x + 4(1) = 7 x + 4 = 7 x = 3 Step 3: Multiply the equations and solve. Multiply the top equation by -4 (-4)(x + 4y = 7) 4x – 3y = 9) y = 1 -4x – 16y = -28 (+) 4x – 3y = 9 -19y = -19

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2) Solve the system using elimination. Step 5: Check your solution. (3, 1) (3) + 4(1) = 7 4(3) - 3(1) = 9 x + 4y = 7 4x – 3y = 9

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What is the first step when solving with elimination? 1. Add or subtract the equations. 2. Multiply the equations. 3. Plug numbers into the equation. 4. Solve for a variable. 5. Check your answer. 6. Determine which variable to eliminate. 7. Put the equations in standard form.

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Which variable is easier to eliminate? 3x + y = 4 4x + 4y = 6 1. x 2. y 3. 6 4. 4

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3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 Step 1: Put the equations in Standard Form. They already are! Step 2: Determine which variable to eliminate. Find the least common multiple of each variable. LCM = 12x, LCM = 12y Which is easier to obtain? Either! I’ll pick y because the signs are already opposite.

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3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 Step 4: Plug back in to find the other variable. 3(1) + 4y = -1 3 + 4y = -1 4y = -4 y = -1 Step 3: Multiply the equations and solve. Multiply both equations (3)(3x + 4y = -1) (4)(4x – 3y = 7) x = 1 9x + 12y = -3 (+) 16x – 12y = 28 25x = 25

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3) Solve the system using elimination. Step 5: Check your solution. (1, -1) 3(1) + 4(-1) = -1 4(1) - 3(-1) = 7 3x + 4y = -1 4x – 3y = 7

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What is the best number to multiply the top equation by to eliminate the x’s? 3x + y = 4 6x + 4y = 6 1. -4 2. -2 3. 2 4. 4

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Solve using elimination. 2x – 3y = 1 x + 2y = -3 1. (2, 1) 2. (1, -2) 3. (5, 3) 4. (-1, -1)

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Find two numbers whose sum is 18 and whose difference 22. 1. 14 and 4 2. 20 and -2 3. 24 and -6 4. 30 and 8

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