The student will be able to: Objective The student will be able to: solve systems of equations using elimination with addition and subtraction. Modified from Skip Tyler PPT

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. You CAN use subtraction, but it is easier to stick to addition. Subtraction causes many errors with double negatives. Elimination is easiest when the equations are in standard form.

Solving a system of equations by elimination using addition. Step 1: Put the equations in Standard Form, but for this purpose A can be negative. Standard Form: Ax + By = C Step 2: Determine which variable to eliminate. Look for variables that have same coefficient(may need to negate an equation to get variables ready). Step 3: Add Down, eliminating one variable. Solve for the variable. Step 4: Plug back in to find the other variable. Substitute the value of the solved variable back into either equation. Step 5: Check your solution. Substitute your ordered pair into BOTH equations.

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

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

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, negate one equation to get opposites. -( 4x + y = 7) becomes -4x – 7 = -7 Now we can ADD DOWN -4x - y = - 7 4x – 2y = -2 -3y = - 9 y = 3 Step 3: Add Down to eliminate x.

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

Solve this system using addition elimination? Try it before you click for answer. You ONLY get better by practicing problems. HINT: Change one of the equations so the 3x’s will cancel. Solve this system using addition elimination? - (3x + y = 4) becomes -3x – y = -4 Now we can ADD DOWN and eliminate the x’s. -3x – y = -4 3x + 4y = 7 3y = 3 , y = 1 Sub back into one of the equations to find x. 3x + 4(1) = 7 3x + 4 = 7 3x = 3 , x=1 Solution ( 1,1) 3x + y = 4 3x + 4y = 7

Solve using elimination. Try it before you click for answer. You ONLY get better by practicing problems. HINT: You can just ADD DOWN to begin. Solve using elimination. 2x – 3y = -2 x + 3y = 17 Now we can ADD DOWN and eliminate the y’s. 3x =15 , x = 5 Sub back into one of the equations to find y. 5 + 3y = 17 3y = 12 , y = 4 Solution ( 5,4) 2x – 3y = -2 x + 3y = 17 (2, 2) (9, 3) (4, 5) (5, 4)

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. Negate one equation to set up your ADD DOWN. - ( 2x + y = 7) becomes -2x – y = -7 Subtract to eliminate y. -2x - y = -7 (+) 4x + y = 5 2x = -2 x = -1 Step 3: ADD DOWN to eliminate the y variable.

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

3) Solve the system using elimination. Try it before you click for answer. You ONLY get better by practicing problems. HINT: You need to get equation(s) into standard form. 3) Solve the system using elimination. 2y -6 = 3x 4x - 8 = 2y Step 1: Put the equations in Standard Form. 3x + 2y = 6 4x - 2y = 8 Step 2: Determine which variable to eliminate. The y’s have an opposite coefficient. ADD DOWN. ADD DOWN to eliminate y. 3x + 2y = 6 4x - 2y = 8 7x = 14 x = 2 Step 3: ADD DOWN to eliminate the y variable.

2) Solve the system using elimination. Equations adjusted to Standard form 3x + 2y = 6 4x – 2y = 8 2y -6 = 3x 4x - 8 = 2y Remember, we just found out that x=2 3(2) + 2y = 6 6 + 2y = 6 2y = 0 , y = 0 Step 4: Plug back in to find the other variable. (2, 0) 3(2) + 2(0) = 6 4(2) - 2(0) = 8 Step 5: Check your solution.