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**Solving Systems Using Elimination**

7.3 Solving Systems Using Elimination

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**7.3 – Solving by Elimination**

Goals / “I can…” Solve systems by adding or subtracting Multiply first when solving systems

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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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**Elimination using Addition**

Consider the system x - 2y = 5 Lets add both equations to each other 2x + 2y = 7 REMEMBER: We are trying to find the Point of Intersection. (x, y)

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**Elimination using Addition**

Consider the system x - 2y = 5 Lets add both equations to each other + 2x + 2y = 7 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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**Elimination using Addition**

Consider the system x - 2y = 5 Lets add both equations to each other + 2x + 2y = 7 3x = 12 x = 4 ANS: (4, y) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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**Elimination using Addition**

Consider the system x - 2y = 5 Lets substitute x = 4 into this equation. 2x + 2y = 7 4 - 2y = 5 Solve for y - 2y = 1 y = 1 2 ANS: (4, y) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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**Elimination using Addition**

Consider the system x - 2y = 5 Lets substitute x = 4 into this equation. 2x + 2y = 7 4 - 2y = 5 Solve for y - 2y = 1 1 2 y = 1 2 ANS: (4, ) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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**Elimination using Addition**

Consider the system 3x + y = 14 4x - y = 7 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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**Elimination using Addition**

Consider the system 3x + y = 14 + 4x - y = 7 7x = 21 x = 3 ANS: (3, y)

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**Elimination using Addition**

Consider the system 3x + y = 14 Substitute x = 3 into this equation 4x - y = 7 3(3) + y = 14 9 + y = 14 y = 5 ANS: (3, ) 5 NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

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**Elimination using Multiplication**

Consider the system 6x + 11y = -5 6x + 9y = -3

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**Elimination using Multiplication**

Consider the system 6x + 11y = -5 + 6x + 9y = -3 12x + 20y = -8 When we add equations together, nothing cancels out

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**Elimination using Multiplication**

Consider the system 6x + 11y = -5 6x + 9y = -3

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**Elimination using Multiplication**

Consider the system -1 ( ) 6x + 11y = -5 6x + 9y = -3

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**Elimination using Multiplication**

Consider the system - 6x - 11y = 5 + 6x + 9y = -3 -2y = 2 y = -1 ANS: (x, ) -1

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**Elimination using Multiplication**

Consider the system 6x + 11y = -5 6x + 9y = -3 Lets substitute y = -1 into this equation y = -1 6x + 9(-1) = -3 6x + -9 = -3 +9 6x = 6 x = 1 ANS: (x, ) -1

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**Elimination using Multiplication**

Consider the system 6x + 11y = -5 6x + 9y = -3 Lets substitute y = -1 into this equation y = -1 6x + 9(-1) = -3 6x + -9 = -3 +9 6x = 6 x = 1 ANS: ( , ) 1 -1

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**Solving a system of equations by elimination using multiplication.**

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

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**1) Solve the system using elimination.**

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

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**1) Solve the system using elimination.**

2x + 2y = 6 3x – y = 5 (2, 1) 2(2) + 2(1) = 6 3(2) - (1) = 5 Step 5: Check your solution. 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! 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) Step 2: Determine which variable to eliminate.

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**2) Solve the system using elimination.**

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

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**2) Solve the system using elimination.**

x + 4y = 7 4x – 3y = 9 (3, 1) (3) + 4(1) = 7 4(3) - 3(1) = 9 Step 5: Check your solution.

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**What is the first step when solving with elimination?**

Add or subtract the equations. Multiply the equations. Plug numbers into the equation. Solve for a variable. Check your answer. Determine which variable to eliminate. Put the equations in standard form.

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**Which variable is easier to eliminate?**

3x + y = 4 4x + 4y = 6 x y 6 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

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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! 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. Step 2: Determine which variable to eliminate.

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**3) Solve the system using elimination.**

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

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**3) Solve the system using elimination.**

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

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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 -4 -2 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

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**Solve using elimination.**

2x – 3y = 1 x + 2y = -3 (2, 1) (1, -2) (5, 3) (-1, -1)

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**Find two numbers whose sum is 18 and whose difference 22.**

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