Solving Simultaneous Equations using Elimination Method

Introduction

• Simultaneous Equations: Pairs of equations that you solve by finding a pair of x and y values working for both.
• Elimination Method: Combines two equations into one, eliminating either x's or y's to solve for one variable.

Steps for Using Elimination Method

1. Label Equations: Number the equations (e.g., equation 1 and equation 2).
2. Combine Equations: Aim to eliminate one variable by combining equations.
3. Solve for One Variable: Use the new equation to solve for the remaining variable.
4. Substitute Back: Use the found value to solve for the other variable in one of the original equations.
5. Verify Solutions: Substitute back the x and y values into the other original equation to ensure correctness.

Example 1

• Equations:
  • Equation 1: ( 7x + 2y = 23 )
  • Equation 2: ( 3x + 2y = 11 )
• Step 1: Subtract equation 2 from equation 1.
  • ( 7x - 3x = 4x )
  • ( 2y - 2y = 0 )
  • ( 23 - 11 = 12 )
  • Result: ( 4x = 12 )
• Step 2: Solve for x.
  • ( x = 3 )
• Step 3: Substitute back to solve for y using equation 1.
  • ( 7(3) + 2y = 23 )
  • ( 21 + 2y = 23 )
  • ( 2y = 2 )
  • ( y = 1 )
• Check: Verify by substituting x and y into equation 2.
  • ( 3(3) + 2(1) = 11 )
  • Confirmed: 9 + 2 = 11

Example 2 (More Complex)

• Equations:
  • Equation 1: ( 4x + y = 10 )
  • Equation 2: ( 3y = 2x - 19 )
• Step 1: Rearrange equation 2.
  • ( -2x + 3y = -19 )
• Step 2: Make x or y terms equal in both equations.
  • Multiply equation 2 by -2: ( 4x - 6y = 38 )
• Step 3: Subtract equation 2 from equation 1.
  • ( 4x - 4x = 0 )
  • ( y + 6y = 7y )
  • ( 10 - 38 = -28 )
  • Result: ( 7y = -28 )
• Step 4: Solve for y.
  • ( y = -4 )
• Step 5: Substitute y back into equation 1 to find x.
  • ( 4x - 4 = 10 )
  • ( 4x = 14 )
  • ( x = 3.5 )
• Check: Verify by substituting x and y into equation 2.
  • ( 3(-4) = 2(3.5) - 19 )
  • Confirmed: -12 = -12

Conclusion

• Verification: Double-check results by substituting back into the alternate equation.
• Resource: Amadeus offers a free learning platform for further practice and progress tracking.
• Additional Resources: Check the video playlist for more lessons.

Additional Information

• Amadeus Platform: Free learning platform for sciences and maths.
• Video Playlist: For more structured learning, videos are arranged in a playlist.