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# Solving Linear Equations

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### Solving Linear Equations

1. 1. Solving Linear Equations Using Graphing Substitution and Elimination Need some homework help? Go to: http://go.hrw.com Keyword: MA1 Homework Help
2. 2. What’s the Deal? <ul><li>There are a number of ways to solve groups of linear equations. </li></ul><ul><li>In this review, we will find points on a coordinate plane that solve linear equations in standard form and y-intercept form. </li></ul>
3. 3. Three Parts <ul><li>Part One – Solve linear equations by graphing. </li></ul><ul><li>Part Two – Solve linear equations by substitution. </li></ul><ul><li>Part Three – Solve linear equations by elimination. </li></ul>
4. 4. Solving for linear equations answers the question: <ul><li>What values of x and y fit into both equations? </li></ul><ul><li>The answer is usually given in (x,y) format (ie. (-4, 6) or (3,8). </li></ul>
5. 5. Remember - Slope intercept form: y = mx + b <ul><li>m = slope </li></ul><ul><li>b = y-intercept </li></ul><ul><li>In y = 1/2x – 7 , the place where the line intercepts the y-axis (called the y-intercept) is negative seven (-7). </li></ul><ul><li>The slope, which is the rise over the run, is ½ (the fraction before the x): </li></ul><ul><ul><li>Rise = up, or plus one (+1) </li></ul></ul><ul><ul><li>Run = right, or plus two (+2). </li></ul></ul>
6. 6. If the slope is ½ <ul><li>Rise </li></ul><ul><li>Run = slope = m </li></ul><ul><li>The rise is 1 and the run is 2. </li></ul><ul><li>From the origin (0,0), go up 1 and right 2. </li></ul>
7. 7. Graphing Systems of equations <ul><li>y = 3x + 1 </li></ul><ul><li>y = -x + 5 </li></ul><ul><li>Since both are in y-intercept format (y=mx+b) find the point through which the line intercepts the y-axis. </li></ul><ul><li>From that point, graph the slope. </li></ul>
8. 8. Graph y=3x + 1 <ul><li>In the following slide, you will see +1 graphed as the y-intercept. </li></ul><ul><li>And the slope rise =3 and run = 1 will be graphed over the y-intercept. </li></ul>
9. 9. y = x + 1 y-intercept
10. 10. Let’s add y = -x + 5 The slope is -1. Or down one And right one.
11. 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. 12. Solve by graphing <ul><li>y = x +3 </li></ul><ul><li>y = x +1 </li></ul><ul><li>The next two slides will show the solution. </li></ul>
13. 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. 14. Now solve equations in standard form. <ul><li>3 x + 2y = -6 and </li></ul><ul><li>-3 x + 2y = 6 </li></ul><ul><li>When graphing, you must convert equations from standard form to y-intercept form. </li></ul><ul><li>Let’s review that from a previous lesson using the equations above… </li></ul>
15. 15. Change 3x + 2y = -6 to y-intercept form <ul><li>3x + 2y = -6 </li></ul><ul><li>- 3x -3x </li></ul><ul><li>2y = -3x - 6 </li></ul><ul><li>Now we need to get y isolated. In this case, let’s divide both sides by 2. </li></ul><ul><li>2y = -3x - 6 </li></ul><ul><li>2 2 2 </li></ul><ul><li>Now simplify. y = - x -3 </li></ul>Subtract -3x from both sides
16. 16. Change -3x + 2y = 6 to y-intercept form <ul><li>-3x + 2y = +6 </li></ul><ul><li>+ 3x 3x </li></ul><ul><li>2y = 3x + 6 </li></ul><ul><li>Get y isolated. Divide both sides by 2. </li></ul><ul><li>2y = 3x + 6 </li></ul><ul><li>2 2 2 </li></ul><ul><li>Now simplify. y = 3 / 2 x + 3 </li></ul>Add 3x to both sides
17. 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. 18. End of Part One
19. 19. Part Two – Solve linear equations by substitution Need Help? Go online to go.hrw.com See your textbook’s “internet connect” notes
20. 20. Here’s what will happen <ul><li>1) Find the x or y value that is isolated, such as x = y+3. The x-value is isolated. </li></ul><ul><li>2) Insert the isolated value into the equation, this allows you to solve for one variable at a time. </li></ul><ul><li>3) Find the value of one variable. </li></ul><ul><li>4) Insert that value into either equation and solve for the second variable. </li></ul>
21. 21. Solve for the following equations. <ul><li>2x + 8y = 1 </li></ul><ul><li>x = 2y </li></ul><ul><li>Step one ( listed on the previous slide ) </li></ul><ul><ul><li>1) Find the x or y value that is isolated, such as x = 2y. The x-value is isolated. </li></ul></ul><ul><ul><li>Since x = 2y, you insert 2y wherever x occurs. </li></ul></ul>
22. 22. Step 2) Insert the isolated value into the equation, this allows you to solve for one variable at a time. <ul><li>2( 2y ) + 8y = 1 </li></ul><ul><li>4y + 8y = 1 </li></ul><ul><li>12y=1 </li></ul><ul><li>12y = 1 </li></ul><ul><li>12 12 </li></ul><ul><li>y = 1 / 12 </li></ul><ul><li>Replace the x with 2y by substitution. </li></ul><ul><li>Multiply 2*2y. </li></ul><ul><li>Combine like terms. </li></ul><ul><li>Divide both sides by 12 </li></ul><ul><li>Solve for y. </li></ul>Careful! You are only half done. You still have to solve for the other variable!
23. 23. Solve for: 2x + 8y = 1 and x = 2y <ul><li>3) Find the value one variable. </li></ul><ul><ul><li>This was done on the previous slide. y = 1 / 12 </li></ul></ul><ul><li>4) Insert that value into either equation and solve for the second variable. </li></ul><ul><ul><li>x = 2 ( ) or </li></ul></ul><ul><li>x = (which is in lowest terms) </li></ul>The solution is (1/6, 1/12)
24. 24. Try one. <ul><li>2x + y = 5 </li></ul><ul><li>x = 7 </li></ul>
25. 25. x is given as 7. Insert 7 for any occurrence of x to solve for y. <ul><li>First solve for y. </li></ul><ul><li>2x + y = 5 </li></ul><ul><li>2(7) + y = 5 </li></ul><ul><li>14 + y = 5 </li></ul><ul><li>-14 -14 </li></ul><ul><li>y = -9 </li></ul><ul><li>Since you know that x = 7 and y = -9, insert those values into a coordinate in (x,y) format. </li></ul><ul><li>Solution: ( 7, -9 ) </li></ul>
26. 26. Solve for x and y. 3x + y = 4 and 5x – 7y = 11 <ul><li>Notice that y can more easily be isolated in the first equation. </li></ul><ul><li>The Plan: </li></ul><ul><ul><li>Let’s isolate y. </li></ul></ul><ul><ul><li>Then we will use the value for y to substitute for y. </li></ul></ul>
27. 27. Subtract 3x from both sides to isolate y. <ul><li>3x + y = 4 </li></ul><ul><li>- 3x -3x </li></ul><ul><li>y = -3x +4 </li></ul><ul><li>5x – 7y = 11 </li></ul><ul><li>5x – 7( -3x +4) = 11 </li></ul><ul><li>Now that we know that y = -3x+4, substitute. </li></ul><ul><li>Which property gets used next? </li></ul>
28. 28. The Distributive Property is used. <ul><li>5x – 7( -3x +4) = 11 </li></ul><ul><li>5x +21x -28 = 11 </li></ul><ul><li>26x – 28 = 11 </li></ul><ul><li>+28 +28 </li></ul><ul><li>Distribute -7 to both terms inside the parentheses. </li></ul><ul><li>-7(-3x) = + 21x </li></ul><ul><li>-7(4) = -28 </li></ul><ul><li>Combine like terms. </li></ul><ul><li>Add 28 to both sides. </li></ul><ul><li>Next: divide both sides by 26. </li></ul>
29. 29. Continuing… <ul><li>26x = 39 </li></ul><ul><li>26 = 26 </li></ul><ul><li>x = 39 / 26 </li></ul><ul><li>x = 3 / 2 </li></ul><ul><li>3( 3 / 2 ) + y = 4 </li></ul><ul><li>Find lowest terms (divide 39 and 26 by the LCM) </li></ul><ul><li>Using the x value, solve for y in one of the equations. </li></ul>
30. 30. 3( 3 / 2 ) + y = 4 <ul><li>3 x 3 + y = 4 </li></ul><ul><li>1 x 2 </li></ul><ul><li>9 / 2 + y = 4 </li></ul><ul><li>- 9 / 2 -4 ½ </li></ul><ul><li>y = -½ </li></ul><ul><li>( 3 / 2 , -½) or (1.5, -0.5) </li></ul><ul><li>Change 3 to a fraction </li></ul><ul><li>Multiply the fractions </li></ul><ul><li>Subtract 9 / 2 [ or 4 ½] from both sides. </li></ul><ul><li>The solutions in fraction and decimal forms. </li></ul>
31. 31. Part Three – Solving equations by Elimination Need some homework help? Go to: http:// go.hrw.com Keyword: MA1 Homework Help
32. 32. 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>
33. 33. 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>
34. 34. 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>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>
35. 35. 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>
36. 36. 2x + 3y = 21 -3x – 3y = -12 You may pick 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>
37. 37. Try One. <ul><li>-4x + 3y = -1 </li></ul><ul><li>4x + 6y = 5 </li></ul>
38. 38. 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>
39. 39. 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>
40. 40. 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>
41. 41. Let’s try 11x + 2y = -8 and 8x + 3y = 5 <ul><li>Our goal is to 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>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. </li></ul>
42. 42. 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>
43. 43. 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>
44. 44. -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>
45. 45. 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>
46. 46. 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>-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>
47. 47. 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>
48. 48. You have reviewed the first three parts of Chapter 7.