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Solving Linear Equations

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  • 1. Solving Linear Equations Using Graphing Substitution and Elimination Need some homework help? Go to: http://go.hrw.com Keyword: MA1 Homework Help
  • 2. What’s the Deal?
    • There are a number of ways to solve groups of linear equations.
    • In this review, we will find points on a coordinate plane that solve linear equations in standard form and y-intercept form.
  • 3. Three Parts
    • Part One – Solve linear equations by graphing.
    • Part Two – Solve linear equations by substitution.
    • Part Three – Solve linear equations by elimination.
  • 4. Solving for linear equations answers the question:
    • What values of x and y fit into both equations?
    • The answer is usually given in (x,y) format (ie. (-4, 6) or (3,8).
  • 5. Remember - Slope intercept form: y = mx + b
    • m = slope
    • b = y-intercept
    • In y = 1/2x – 7 , the place where the line intercepts the y-axis (called the y-intercept) is negative seven (-7).
    • The slope, which is the rise over the run, is ½ (the fraction before the x):
      • Rise = up, or plus one (+1)
      • Run = right, or plus two (+2).
  • 6. If the slope is ½
    • Rise
    • Run = slope = m
    • The rise is 1 and the run is 2.
    • From the origin (0,0), go up 1 and right 2.
  • 7. Graphing Systems of equations
    • y = 3x + 1
    • y = -x + 5
    • Since both are in y-intercept format (y=mx+b) find the point through which the line intercepts the y-axis.
    • From that point, graph the slope.
  • 8. Graph y=3x + 1
    • In the following slide, you will see +1 graphed as the y-intercept.
    • And the slope rise =3 and run = 1 will be graphed over the y-intercept.
  • 9. y = x + 1 y-intercept
  • 10. Let’s add y = -x + 5 The slope is -1. Or down one And right one.
  • 11. The coordinates of the intersecting point is your solution. The lines inter- cept at (1, 4) so the solution is x=1, y =4. The lines inter- cept at (1, 4) so the solution is x=1, y =4.
  • 12. Solve by graphing
    • y = x +3
    • y = x +1
    • The next two slides will show the solution.
  • 13. The coordinates of the intersecting point is your solution. The lines inter- cept at (-20,-12) so the solution is x= -20, y = -12.
  • 14. Now solve equations in standard form.
    • 3 x + 2y = -6 and
    • -3 x + 2y = 6
    • When graphing, you must convert equations from standard form to y-intercept form.
    • Let’s review that from a previous lesson using the equations above…
  • 15. Change 3x + 2y = -6 to y-intercept form
    • 3x + 2y = -6
    • - 3x -3x
    • 2y = -3x - 6
    • Now we need to get y isolated. In this case, let’s divide both sides by 2.
    • 2y = -3x - 6
    • 2 2 2
    • Now simplify. y = - x -3
    Subtract -3x from both sides
  • 16. Change -3x + 2y = 6 to y-intercept form
    • -3x + 2y = +6
    • + 3x 3x
    • 2y = 3x + 6
    • Get y isolated. Divide both sides by 2.
    • 2y = 3x + 6
    • 2 2 2
    • Now simplify. y = 3 / 2 x + 3
    Add 3x to both sides
  • 17. Graph the equations: y = - 3 / 2 x -3 and y = 3 / 2 x + 3 x = 2, y = 0 The solution is (2,0)
  • 18. End of Part One
  • 19. Part Two – Solve linear equations by substitution Need Help? Go online to go.hrw.com See your textbook’s “internet connect” notes
  • 20. Here’s what will happen
    • 1) Find the x or y value that is isolated, such as x = y+3. The x-value is isolated.
    • 2) Insert the isolated value into the equation, this allows you to solve for one variable at a time.
    • 3) Find the value of one variable.
    • 4) Insert that value into either equation and solve for the second variable.
  • 21. Solve for the following equations.
    • 2x + 8y = 1
    • x = 2y
    • Step one ( listed on the previous slide )
      • 1) Find the x or y value that is isolated, such as x = 2y. The x-value is isolated.
      • Since x = 2y, you insert 2y wherever x occurs.
  • 22. Step 2) Insert the isolated value into the equation, this allows you to solve for one variable at a time.
    • 2( 2y ) + 8y = 1
    • 4y + 8y = 1
    • 12y=1
    • 12y = 1
    • 12 12
    • y = 1 / 12
    • Replace the x with 2y by substitution.
    • Multiply 2*2y.
    • Combine like terms.
    • Divide both sides by 12
    • Solve for y.
    Careful! You are only half done. You still have to solve for the other variable!
  • 23. Solve for: 2x + 8y = 1 and x = 2y
    • 3) Find the value one variable.
      • This was done on the previous slide. y = 1 / 12
    • 4) Insert that value into either equation and solve for the second variable.
      • x = 2 ( ) or
    • x = (which is in lowest terms)
    The solution is (1/6, 1/12)
  • 24. Try one.
    • 2x + y = 5
    • x = 7
  • 25. x is given as 7. Insert 7 for any occurrence of x to solve for y.
    • First solve for y.
    • 2x + y = 5
    • 2(7) + y = 5
    • 14 + y = 5
    • -14 -14
    • y = -9
    • Since you know that x = 7 and y = -9, insert those values into a coordinate in (x,y) format.
    • Solution: ( 7, -9 )
  • 26. Solve for x and y. 3x + y = 4 and 5x – 7y = 11
    • Notice that y can more easily be isolated in the first equation.
    • The Plan:
      • Let’s isolate y.
      • Then we will use the value for y to substitute for y.
  • 27. Subtract 3x from both sides to isolate y.
    • 3x + y = 4
    • - 3x -3x
    • y = -3x +4
    • 5x – 7y = 11
    • 5x – 7( -3x +4) = 11
    • Now that we know that y = -3x+4, substitute.
    • Which property gets used next?
  • 28. The Distributive Property is used.
    • 5x – 7( -3x +4) = 11
    • 5x +21x -28 = 11
    • 26x – 28 = 11
    • +28 +28
    • Distribute -7 to both terms inside the parentheses.
    • -7(-3x) = + 21x
    • -7(4) = -28
    • Combine like terms.
    • Add 28 to both sides.
    • Next: divide both sides by 26.
  • 29. Continuing…
    • 26x = 39
    • 26 = 26
    • x = 39 / 26
    • x = 3 / 2
    • 3( 3 / 2 ) + y = 4
    • Find lowest terms (divide 39 and 26 by the LCM)
    • Using the x value, solve for y in one of the equations.
  • 30. 3( 3 / 2 ) + y = 4
    • 3 x 3 + y = 4
    • 1 x 2
    • 9 / 2 + y = 4
    • - 9 / 2 -4 ½
    • y = -½
    • ( 3 / 2 , -½) or (1.5, -0.5)
    • Change 3 to a fraction
    • Multiply the fractions
    • Subtract 9 / 2 [ or 4 ½] from both sides.
    • The solutions in fraction and decimal forms.
  • 31. Part Three – Solving equations by Elimination Need some homework help? Go to: http:// go.hrw.com Keyword: MA1 Homework Help
  • 32. 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?
  • 33. 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
  • 34. 2x + 3y = 21 -3x – 3y = -12
    • Notice that the coefficients with the y value are opposites. (+3 and -3).
    • 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.
  • 35. 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.
  • 36. 2x + 3y = 21 -3x – 3y = -12 You may pick 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
  • 37. Try One.
    • -4x + 3y = -1
    • 4x + 6y = 5
  • 38. 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…
  • 39. 4x = 21 / 9
    • Divide both sided by 4.
    • 4x = 21 / 9
    • 4 4
    • X =
    • X =
    • To divide fractions, multiply by the reciprocal
  • 40. Ready to go one more step?
    • What if you don’t have an easy choice.
    • You may find that neither equation has opposite coefficients.
  • 41. Let’s try 11x + 2y = -8 and 8x + 3y = 5
    • Our goal is to eliminate a variable using opposite coefficients.
    • It looks like we should use 2y and 3y since they are smaller numbers.
    • If we multiply both sides of the top equation by -3 and both sides of the bottom by 2, we should get coefficients of 6 and -6.
  • 42. 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.
  • 43. 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
  • 44. -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)
  • 45. 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…
  • 46. One more for practice
    • 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)
  • 47. 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.
  • 48. You have reviewed the first three parts of Chapter 7.

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