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 and also whether to Multiply. 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.

Solve: by ELIMINATION x + y = 4 2x + 3y = 9 Like variables must be lined under each other. Solve: by ELIMINATION x + y = 4 2x + 3y = 9 We need to eliminate (get rid of) a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2.

( ) -2 Y = 1 THEN---- X + Y = 4 -2X - 2 Y = - 8 2X + 3Y = 9 Now add the two equations and solve. THEN----

Now check our answers in both equations------ Substitute your answer into either original equation and solve for the second variable. X + Y = 4 X + 1 = 4 - 1 -1 X = 3 (3,1) Answer Now check our answers in both equations------

x + y = 4 3 + 1 = 4 4 = 4 2x + 3y = 9 2(3) + 3(1) = 9 6 + 3 = 9 9 = 9

5x + 3y = 11 5x = 2y + 1 EXAMPLE #1: STEP1: Write both equations in Ax + By = C form. 5x + 3y =1 5x - 2y =1 STEP 2: Use subtraction to eliminate 5x. 5x + 3y =11 5x + 3y = 11 -(5x - 2y =1) -5x + 2y = -1 Note: the (-) is distributed. STEP 3: Solve for the variable. 5x + 3y =11 -5x + 2y = -1 5y =10 y = 2

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).

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

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.

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 = 10 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

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).

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

NOW solve these using elimination: 1. 2. 2x + 4y =1 x - 4y =5 2x – y =6 x + y = 3

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.

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

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

Using Elimination to Solve a Word Problem: x + y = 180 x =10 + 3y x + 42.5 = 180 x = 180 - 42.5 x = 137.5 (137.5, 42.5) x + y = 180 -(x - 3y = 10) 4y =170 y = 42.5

Using Elimination to Solve a Word Problem: The sum of two numbers is 70 and their difference is 24. Find the two numbers.

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

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

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)

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 114. Find the numbers.