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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
• 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.