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**3-2: Solving Systems of Equations using Elimination**

Steps: 1. Place both equations in Standard Form, Ax + By = C. 2. Determine which variable to eliminate with Addition or Subtraction. 3. Solve for the variable left. 4. Go back and use the found variable in step 3 to find second variable. 5. Check the solution in both equations of the system.

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5x + 3y = 11 5x = 2y + 1 EXAMPLE #1: STEP1: Write both equations in Ax + By = C form x + 3y =1 5x - 2y =1 STEP 2: Use subtraction to eliminate 5x x + 3y = x + 3y = 11 -(5x - 2y =1) x + 2y = -1 Note: the (-) is distributed. STEP 3: Solve for the variable. 5x + 3y =11 -5x + 2y = -1 5y = y = 2

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**The solution to the system is (1,2).**

5x + 3y = 11 5x = 2y + 1 STEP 4: Solve for the other variable by substituting into either equation. 5x + 3y =11 5x + 3(2) =11 5x + 6 =11 5x = 5 x = 1 The solution to the system is (1,2).

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**5x + 3y = 11 5x = 2y + 1 5(1) + 3(2) =11 5(1) = 2(2) + 1 5 + 6 =11**

Step 5: Check the solution in both equations. The solution to the system is (1,2). 5x + 3y = 11 5(1) + 3(2) =11 5 + 6 =11 11=11 5x = 2y + 1 5(1) = 2(2) + 1 5 = 4 + 1 5=5

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**Solving Systems of Equations using Elimination**

Steps: 1. Place both equations in Standard Form, Ax + By = C. 2. Determine which variable to eliminate with Addition or Subtraction. 3. Solve for the remaining variable. 4. Go back and use the variable found in step 3 to find the second variable. 5. Check the solution in both equations of the system.

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Example #2: x + y = 10 5x – y = 2 Step 1: The equations are already in standard form: x + y = 10 5x – y = 2 Step 2: Adding the equations will eliminate y. x + y = x + y = 10 +(5x – y = 2) +5x – y = +2 Step 3: Solve for the variable. x + y = 10 +5x – y = +2 6x = 12 x = 2

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**Solution to the system is (2,8).**

x + y = 10 5x – y = 2 Step 4: Solve for the other variable by substituting into either equation. x + y = 10 2 + y = 10 y = 8 Solution to the system is (2,8).

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**x + y =10 5x – y =2 2 + 8 =10 5(2) - (8) =2 10 – 8 =2 10=10 2=2**

Step 5: Check the solution in both equations. Solution to the system is (2,8). x + y =10 2 + 8 =10 10=10 5x – y =2 5(2) - (8) =2 10 – 8 =2 2=2

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**NOW solve these using elimination:**

1. 2. 2x + 4y =1 x - 4y =5 2x – y =6 x + y = 3

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Solve: 3s - 2t = s + t = 6 If there isn’t a quick cancel, you have to make one! We could multiply the second equation by 2 and the t terms would be inverses. OR We could multiply the first equation by 4 and the second equation by -3 to make the s terms inverses. Let’s multiply the second equation by 2 to eliminate t. (It’s easier.) 3s - 2t = s – 2t = 10 2(4s + t = 6) s + 2t = 12 Add and solve: s + 0t = 22 11s = 22 s = 2 Insert the value of s to find the value of t (2) - 2t = t = -2 The solution is (2, -2).

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**We do one… you do one: 1) 5x + 4y = 12 2) 5m + 2n = -8**

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**Using Elimination to Solve a Word Problem:**

Two angles are supplementary. The measure of one angle is 10 degrees more than three times the other. Find the measure of each angle.

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**Using Elimination to Solve a Word Problem:**

Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle. x = degree measure of angle #1 y = degree measure of angle #2 Therefore x + y = 180

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**Using Elimination to Solve a Word Problem:**

Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle. x + y = 180 x =10 + 3y

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**Using Elimination to Solve a Word Problem:**

x + y = 180 x =10 + 3y x = 180 x = x = 137.5 (137.5, 42.5) x + y = 180 -(x - 3y = 10) 4y =170 y = 42.5

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**Using Elimination to Solve a Word Problem:**

The sum of two numbers is 70 and their difference is 24. Find the two numbers.

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**Using Elimination to Solve a Word problem:**

The sum of two numbers is 70 and their difference is 24. Find the two numbers. x = first number y = second number Therefore, x + y = 70

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**Using Elimination to Solve a Word Problem:**

The sum of two numbers is 70 and their difference is 24. Find the two numbers. x + y = 70 x – y = 24

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**Using Elimination to Solve a Word Problem:**

x + y =70 x - y = 24 47 + y = 70 y = 70 – 47 y = 23 2x = 94 x = 47 (47, 23)

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**Now you Try to Solve These Problems Using Elimination.**

Find two numbers whose sum is 18 and whose difference is 22. The sum of two numbers is 128 and their difference is Find the numbers.

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