Solving Equations And Formulas


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solving multivariable equations for a specific variable (algebra 1)

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Solving Equations And Formulas

  1. 1. Solving Equations and Formulas Chapter 3 section 8
  2. 2. Solving for a specific Variable <ul><li>Sometimes equations involve multiple variables, or letters that stand for specific things, such as scientific formulas and physics equations </li></ul>
  3. 3. Geometry Formulas: <ul><li>The formula for Circumference of a circle is: </li></ul><ul><li>C =2 πr (2 x pi x radius) </li></ul><ul><li>We can solve the formula for r: (this means get r by itself!) </li></ul>
  4. 4. What do you think we could do? <ul><li>If the equation is set up as C = 2 π r, and you are asked to get r by itself, then you need to first ask yourself: What is attached to r? How can you “undo” the operation? </li></ul><ul><li>Since 2 π is “stuck” to r by multiplication, you could get rid of it by division…… </li></ul>
  5. 5. Like this! <ul><li>C = 2 π r </li></ul><ul><li>Divide both sides by 2 π: 2 π 2 π </li></ul><ul><li>This leaves you with the following: </li></ul><ul><li> C = r </li></ul><ul><li>2 π </li></ul><ul><li>That’s it! That is all there is to it! </li></ul><ul><li>Now r is by itself, and all of the other “stuff” is on the other side! </li></ul>
  6. 6. Here’s another one: <ul><li>Try solving the following formula for x: </li></ul><ul><li>5x + y = x </li></ul>
  7. 7. Follow these steps to get x: <ul><li>5x + y = x </li></ul><ul><li>Move your x’s together, to the same side: Which one should you move? The 5x or the x??? </li></ul><ul><li>5x + y = x </li></ul><ul><li>- 5x -5x </li></ul><ul><li>(you should move the 5x because it needs to be with the x on the other side, and it needs to be away from the y! </li></ul>
  8. 8. Steps 2 & 3: <ul><li>Now your equation should look like this! </li></ul><ul><li>y = -4x </li></ul><ul><li>2. What would be the next thing you should do to get the x by itself? </li></ul><ul><li>(If you said DIVIDE BY – 4, THEN YOU’RE RIGHT!) </li></ul><ul><li> y = -4x </li></ul><ul><li>-4 -4 </li></ul><ul><li>3. Your answer will be y = x </li></ul><ul><li>-4 </li></ul>
  9. 9. How ‘bout another one?!? <ul><li>Solve 6 – ay = 4( a – b ) for a </li></ul>
  10. 10. Just follow the steps you have been using in other types of equations! <ul><li>6 – ay = 4( a – b ) </li></ul><ul><li>Distributive property: </li></ul><ul><li> 6 – ay = 4 a – 4 b </li></ul><ul><li>Get the things with an “a” together, and the things that don’t have an “a” together: </li></ul><ul><li>6 – ay = 4 a – 4 b </li></ul><ul><li>- 4a - 4a </li></ul><ul><li>Simplify to get: </li></ul><ul><li> 6 – ay – 4a = - 4b </li></ul>
  11. 11. Big Finish: <ul><li>6 – ay – 4a = - 4b </li></ul><ul><li>-6 -6 </li></ul><ul><li>Simplify to get: - ay – 4a = - 4b – 6 </li></ul><ul><li>Now, here’s the tricky part: You have two things on the left that have a’s and you can’t combine them because they aren’t like terms! </li></ul><ul><li>You have to do something called Factoring: it goes like this: Ask yourself what a is being multiplied by in the two terms on the left: there is an a with – y, and an a with – 4, so you group them in parentheses: </li></ul>
  12. 12. Factor: <ul><li>- ay – 4a = - 4b – 6 </li></ul><ul><li>Now looks like this: </li></ul><ul><li>a(- y – 4) = - 4b – 6 </li></ul><ul><li>(Its like the distributive property, backwards!) </li></ul><ul><li>To get the a by itself, divide both sides by the stuff in the parentheses: </li></ul><ul><li>a(- y – 4) = - 4b – 6 </li></ul><ul><li>(- y – 4) (- y – 4) </li></ul><ul><li>Your answer looks REALLY WEIRD!, but that’s OK! </li></ul>
  13. 13. Your answer should look like this: <ul><li>a= - 4b – 6 </li></ul><ul><li>- y – 4 </li></ul><ul><li>(you don’t have to keep the () on bottom any more) </li></ul><ul><li>How’s that </li></ul><ul><li>for fun!?! </li></ul>
  14. 14. Try one more : (if your brain is not already fried!) <ul><li>The perimeter of a square field is given by the equation P = 2 l + 2 w , where P represents the </li></ul><ul><li>perimeter, l represents the length of the field, and w represents the width of the field. </li></ul><ul><li>Solve the formula for l . </li></ul>
  15. 15. What should you do first?
  16. 16. Answer: <ul><li>P = 2 l + 2 w </li></ul><ul><li>- 2w - 2w </li></ul><ul><li>P - 2w = 2l </li></ul><ul><li>2 2 </li></ul><ul><li>P - 2w = l </li></ul><ul><li>2 </li></ul>
  17. 17. OMG!!!! <ul><li>Yes, these are hard! </li></ul><ul><li>Yes, you have to do them! </li></ul><ul><li>Yes, you have an assignment! </li></ul><ul><li>Book page 168 14 – 32 even </li></ul><ul><li>(10 problems) </li></ul>
  18. 18. You’ll thank me one day…….