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7 3elimination

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  • 1. Part Three – Solving equations by Elimination Holt Algebra I Text pages 330-334
  • 2. Solve using substitution after manipulating equations in standard form.
    • 2x + 3y = 21
    • -3x – 3y = -12
    • Which value, x or y, should we work with first?
    • This looks like a very long, drawn-out problem. Is there a better way?
  • 3. Let’s solve by elimination.
    • This method uses opposites to eliminate one of the variables.
    • Which variable should be eliminated?
    • 2x + 3y = 21
    • -3x – 3y = -12
  • 4. 2x + 3y = 21 -3x – 3y = -12
    • Notice that the coefficients with the y value are opposites. (+3 and -3).
    • Use Columns
    • Solve for remaining variable.
    • Substitute that value.
    • If we combine these two equations together in columns, we can eliminate the y values.
    • We will solve for x and then insert it’s value into one of the original equations to solve for y.
  • 5. The steps and explanations
    • 2x + 3y = 21 -3x – 3y = -12 -1x + 0 = 9
    • -1x + 0= 9
    • -1 -1
    • x = -9
    • Add terms from top to bottom.
      • +2x - 3x
      • +3y - 3y
    • Divide both sides by -1 .
    • Now go back and insert -9 for x.
  • 6. 2x + 3y = 21 -3x – 3y = -12 You may insert (x= -9) into either one.
    • 2(-9) + 3y = 21
    • -18 + 3y = 21
    • (add 18 to both sides)
    • +3y = 39
    • 3 3
    • y = 13
    • Solution (-9, 13)
    • -3(-9) – 3y = -12
    • +27 – 3y = -12
    • (subtract 27 from both sides)
    • -3y = -39
    • -3 -3
    • y = 13
  • 7. Try One.
    • -4x + 3y = -1
    • 4x + 6y = 5
  • 8. Eliminate the x values.
    • -4x + 3y = -1
    • 4x + 6y = 5
    • 9y = 4
    • 9y = 4
    • 9 9
    • y = 4 / 9
    • Solve for x.
    • 4x + 6( 4 / 9 ) = 5
    • 4x + 24 / 9 = 45 / 9
    • Subtract 21 / 9 from both sides.
    • 4x = 2 1 / 3
    • Go to the next slide…
  • 9. 4x = 21 / 9
    • Divide both sided by 4.
    • 4x = 21 / 9
    • 4 4
    • x =
    • x =
    • To divide fractions, multiply by the reciprocal
  • 10. Ready to go one more step?
    • What if you don’t have an easy choice.
    • You may find that neither equation has opposite coefficients.
    11x + 2y = -8 8x + 3y = 5
  • 11. Let’s try 11x + 2y = -8 and 8x + 3y = 5
    • Goal
    • Observe
    • Multiply
    • eliminate a variable using opposite coefficients.
    • It looks like we should use 2y and 3y since they are smaller numbers
    • both sides of the top equation by -3 and both sides of the bottom by 2, we should get coefficients of 6 and -6.
  • 12. Multiply both sides
    • (11x + 2y) = (-8) (8x + 3y) = (5)
    • -3(11x + 2y) = (-8)-3
    • 2(8x + 3y) = (5)2
    • We’ll put all four values into parentheses.
    • Multiply both sides of the top by -3
    • Multiply both sides of the second equation by 2.
  • 13. Results of the First Steps
    • -3(11x + 2y) = (-8)-3
    • 2(8x + 3y) = (5)2
    • -----------------------
    • -33x – 6y = +24
    • 16x + 6y = +10
    • -17x + 0 = 34
    • From the previous slide
    • Use the distributive property
    • Now eliminate
  • 14. From previous slide -17x = 34 x = -2
    • 11x + 2y = -8
    • 11(-2) + 2y = -8
    • -22+ 2y = -8
    • 2y = 14
    • y = 7
    • Pick one of the original equations.
    • Solve for the other variable.
    • Add 22 to both sides. -8 +22 = 14.
    • Solution (-2, 7)
  • 15. One more for practice
    • 3x - 2y = 2
    • 4x – 7y = 33
    • --------------------
    • -4(3x - 2y) = (2)-4
    • 3(4x – 7y) = (33)3
    • -------------------------
    • Solution on the next slide…
  • 16. One more for practice - Solution
    • 3x - 2y = 2
    • 4x – 7y = 33
    • --------------------
    • -4(3x - 2y) = (2)-4
    • 3(4x – 7y) = (33)3
    • -------------------------
    • -12x + 8y = -8
    • 12x – 21y = 99
    • -----------------------
    • -13y = 91
    • -13y = 91
    • -13 -13
    • y= -7
    • ---------------------------
    • 3x-2(-7)= 2
    • 3x + 14 = 2
    • 3x = -12
    • x= -4
    • -------------
    • Solution (-4, -7)
  • 17. Which way of solving works best for you?
    • Graphing?
    • Substitution?
    • Elimination?
    • Make sure you know them all in order to pick the best way to solve each problem.
  • 18. Assignment: 334:15-33 & 41-47 odds