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7.3

7.3. Solving Systems Using Elimination. 7.3 – Solving by Elimination. Goals / “I can…” Solve systems by adding or subtracting Multiply first when solving systems. Solving Systems of Equations.

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7.3

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  1. 7.3 Solving Systems Using Elimination

  2. 7.3 – Solving by Elimination • Goals / “I can…” • Solve systems by adding or subtracting • Multiply first when solving systems

  3. 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…

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

  5. + 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.

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

  7. 1 y = 2 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  ANS: (4, y) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  8. 1 2 1 2 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 =  ANS: (4, ) NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.

  9. 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.

  10. + Elimination using Addition Consider the system 3x + y = 14 4x - y = 7 7x = 21 x = 3  ANS: (3, y)

  11. 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.

  12. Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3

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

  14. Elimination using Multiplication Consider the system 6x + 11y = -5 6x + 9y = -3

  15. Elimination using Multiplication Consider the system -1 ( ) 6x + 11y = -5 6x + 9y = -3

  16. Elimination using Multiplication Consider the system - 6x - 11y = 5 + 6x + 9y = -3 -2y = 2  y = -1 ANS: (x, ) -1

  17. +9 +9 Elimination using Multiplication Consider the system 6x + 11y = -5 Lets substitute y = -1 into this equation 6x + 9y = -3 y = -1 6x + 9(-1) = -3 6x + -9 = -3 6x = 6  x = 1 -1 ANS: (x, )

  18. +9 +9 Elimination using Multiplication Consider the system 6x + 11y = -5 Lets substitute y = -1 into this equation 6x + 9y = -3 y = -1 6x + 9(-1) = -3 6x + -9 = -3 6x = 6  x = 1 -1 ANS: ( , ) 1

  19. Solving a system of equations by elimination using multiplication. Standard Form: Ax + By = C Step 1: Put the equations in Standard Form. 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.

  20. 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 multiplythe bottom equation by 2) Step 2: Determine which variable to eliminate.

  21. 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.

  22. 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.

  23. 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 multiplythe top equation by -4 to make them inverses) Step 2: Determine which variable to eliminate.

  24. 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.

  25. 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.

  26. 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.

  27. Which variable is easier to eliminate? • x • y • 6 • 4 3x + y = 4 4x + 4y = 6

  28. 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.

  29. 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.

  30. 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.

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

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

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

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