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