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  1. Today’s Agenda • E - How can graphs be used to solve systems of linear equations? • A - Warm-up: Bell Ringer Patterns B • Review: Graph linear equations from slope-intercept form (graphic organizer) • T - New: System of Linear Equations (graphic organizer and video) • Class work: A Good Deal Shoulder Buddy Practice Problem • S – 3-2-1 Homework: Graphing Equations worksheet #1-6 Smyrna High School Geometry CP

  2. Warm-up Bell Ringer Day 1~Discovering Patterns… • Complete the pattern with your group. Then answer the questions as best you can. Smyrna High School Geometry CP

  3. Pattern B • Draw the penny pattern in figure 4. • How many pennies are in figure 5? • How many pennies are in figure 10? • What figure has 63 pennies? • Write a rule for this pattern? Smyrna High School Geometry CP

  4. Unit 1 – Lesson 3: Systems of Linear Equations Systems of Equations Two or more linear equations that have to be solved simultaneously. Smyrna High School Geometry CP

  5. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • In the slope-intercept form of a linear equation, each letter represents a special value. • m = the slope of the equation (steepness). • b = the y-intercept (starting number). • With these two facts, an equation in slope-intercept form can be graphed. Smyrna High School Geometry CP

  6. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • With these two facts, lets look at some examples. • y = 4x + 2 • First locate point b on the graph and mark it. Since this is the point where the graph of the equation crosses the y-axis, the x value will be zero and the y value will be b in the equation (0, b). Smyrna High School Geometry CP

  7. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Second, determine the slope (m) of the equation. • What number is before the x in the equation? • If there is not a number in front of the letter x, remember the number 1 is understood. Smyrna High School Geometry CP

  8. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Slope is commonly know as the rise/run. • The slope of the example can be written as 4/1. • Use these values to plot some more points. Smyrna High School Geometry CP

  9. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Start at the y-intercept (0, 2). • From this point use the slope of the equation to find a second point. • The slope of the line is 4/1 so count up 4 spaces (the rise) and count to the right 1 space (the run). • Repeat these steps to find a third point. Smyrna High School Geometry CP

  10. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Remember, the slope of the line will determine the direction of the rise and run. • Going up is a positive direction. Then run to the right. • Going down is a negative direction. Then run to the right. Smyrna High School Geometry CP

  11. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Remember, a negative divided by a negative is a positive. • Positive slopes will start lower on the left side of the graph and end up higher on the right side of the graph. • Negative slopes will start higher on the left side of the graph and end up lower on the right side of the graph. Smyrna High School Geometry CP

  12. Pattern C • How many toothpicks are in figure 5? • How many toothpicks are in figure 10? • What figure has 44 pennies? • Write a rule for this pattern. Smyrna High School Geometry CP

  13. Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • For practice, graph the following equations in slope-intercept form. • y = -x + 6 • y = 2x – 4 • y = ¼ x Smyrna High School Geometry CP

  14. -1/1 (0, 6) 2/1 (0, - 4) (0, 0) 1/4 Smyrna High School Geometry CP

  15. Graph! • http://www.youtube.com/watch?v=2BHzXItkByU Smyrna High School Geometry CP

  16. Ticket In The Door Recap • For your equation do the following: • Circle & Identify the y-intercept (don’t forget to circle the sign also) • Graph the y-intercept (always on the y-axis) • Identify the slope (write it as a fraction) • Use slope to plot more points (ryse/run) • Draw a line through the points You may use your “yellow” graphic organizer to help guide you through the steps! Smyrna High School Geometry CP

  17. Smyrna High School Geometry CP

  18. Homework check • Use the red pen to make corrections to your homework • Now, is NOT the time to do your homework • I should not see any writing utensils except the red pen • If you didn’t do your homework then you can turn it in tomorrow for half credit kuta graphing lines in slope intercept form hw.pdf Smyrna High School Geometry CP

  19. Systems of Equations • Let’s take a look at a few examples. • Your task: I Notice, I Wonder. • Discuss and write down some things that you notice about the examples. • Discuss and write down some things that you wonder about the examples. Smyrna High School Geometry CP

  20. Systems of Equations: Example 1 Smyrna High School Geometry CP

  21. Systems of Equations: Example 2 Smyrna High School Geometry CP

  22. Systems of Equations: Example 3 Smyrna High School Geometry CP

  23. Graphing Method Substitution Method Elimination Method 3 Methods to Solve System of Equations Smyrna High School Geometry CP

  24. 1 Exact Solution No Solution Infinitely Many Solutions 3 Possible Solution Types Smyrna High School Geometry CP

  25. 1 Exact Solution Point of intersection Smyrna High School Geometry CP

  26. Another type of solution • How would you describe these lines? Y = 3x + 2 Y = 3x - 4 What do you think the solution (point of intersection) is? Smyrna High School Geometry CP

  27. No Solution Parallel Lines Will not intersect same slope different y-intercepts Smyrna High School Geometry CP

  28. Another type of solution • What do you notice about the graphs and equations? Y = -3x + 4 3x + y = 4 What do you think the solution (point of intersection) is? Smyrna High School Geometry CP

  29. Infinitely Many Solutions SAME LINE They are the SAME equation (just written in different forms) They have same slope and same y-intercept. Smyrna High School Geometry CP

  30. The Goal of Solving Systems • To find one pair (x, y) of values that satisfies both linear equations. • The one pair of values that makes both equations true. Smyrna High School Geometry CP

  31. Hamilton High School Smyrna High School Geometry CP

  32. Hamilton High School • Students in the Hamilton High School science club want to raise $240 to buy a special eyepiece for the high-powered telescope at their school. The PTA offered to pay club members for an after-school work project that would clean up a nearby park and recreation center building. • Because the outdoor work was harder and dirtier, the deal with the PTA would pay $20 for each outdoor worker ad $10 for each indoor worker. The club had 18 members eager to work on the project. Smyrna High School Geometry CP

  33. Hamilton High School • What do the variables x and y represent in these equations? • What problem condition is represented by the equation 20x+ 10y = 240? • What problem condition is represented by the equation x+ y = 18? • Determine 3 combinations of numbers of outdoor and indoor workers that will allow the club to earn just enough money to buy the telescope eyepiece? Smyrna High School Geometry CP

  34. Think-Pair-ShareLet’s Look at the Solution Complete Problem 1 - Parts a, b, c & d. Try it on your own Think: 5 minutes Check with a partner Pair: 2 minutes Whole group discussion Share: 5 minutes Smyrna High School Geometry CP

  35. Hamilton High School • What does the x represent? x: # of outdoor workers • What does the y represent? • y: # of indoor workers Smyrna High School Geometry CP

  36. Hamilton High School 20x + 10y = 240 • What does this equation represent in the problem? • 20x + 10y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor workers Smyrna High School Geometry CP

  37. Hamilton High School x + y = 18 • What does this equation represent in the problem? • x + y = 18 shows that the # of club members who will work Smyrna High School Geometry CP

  38. Let’s finish the problem • Now, take a look at your 3 combination of workers from part d. • Will any of the combinations also put each willing club member to work? Smyrna High School Geometry CP

  39. Smyrna High School Geometry CP

  40. Today’s Agenda • E - How can graphs be used to solve systems of linear equations? • A - Warm-up: Bell Ringer Patterns • Review: Graph linear equations from slope-intercept form (graphic organizer) • T - New: Graph linear equations from intercepts (graphic organizer) • Class work: A Good Deal Shoulder Buddy Practice Problem, Verifying solutions, Solving systems • S – Journal entry Homework: Pattern D Smyrna High School Geometry CP

  41. WARMUP: Pattern E • Draw the pattern for figure 5 • How many white tiles are in figure 5. • How many white tiles are in figure 8? • What figure has 72 white tiles. • Write a rule for the pattern. Smyrna High School Geometry CP

  42. HOMEWORK: Pattern D • How many toothpicks are in figure 5? • How many toothpicks are in figure 10? • What figure has 44 pennies? • Write a rule for this pattern. Smyrna High School Geometry CP

  43. How Do We Graph a Linear Equation??? One way to graph a linear equation is in the form y = mx + b, where m is the slope and b is the y-intercept Smyrna High School Geometry CP

  44. How Do We Graph a Linear Equation not in y=mx+b form??? Let’s Practice: -3x + 2y = 12 • Since it is not in y = mx + b, we will find the x- and y-intercepts and graph those points. • Find the x-intercept: Let y = 0 and solve the equation for x. • Find the y-intercept: Let x = 0 and solve the equation for y. Smyrna High School Geometry CP

  45. How Do We Graph a Linear Equation??? Let’s Practice: -3x + 2y = 12 • Find the x-intercept: Let y = 0 and solve the equation for x. -3x + 2(0) = 12 -3x = 12 -3 -3 x = -4 or (-4, 0) Smyrna High School Geometry CP

  46. How Do We Graph a Linear Equation??? Let’s Practice: -3x + 2y = 12 • Find the y-intercept: Let x = 0 and solve the equation for y. -3(0) + 2y= 12 2y= 12 2 2 y = 6 or (0, 6) Smyrna High School Geometry CP

  47. How Do We Graph a Linear Equation??? You try… 6x - 4y = 12 Smyrna High School Geometry CP

  48. Time to practice… • Let’s do #5 & 6 together • Now it is your turn to finish the odd problems (1, 3, 7) Make sure that you have me check these problems (10 mins) • If you finish early, go ahead and finish the even problems (2, 4, 8). • If you don’t finish the evens in class, you can finish them for homework. graphing lines using intercepts.pdf Smyrna High School Geometry CP

  49. How Do We Graph a Linear Equation??? Let’s Practice with Hamilton High: 20x + 10y = 240 • Find the x-intercept: Let y = 0 and solve the equation for x. • 20x + 10(0) = 240 • 20x = 240 • 20 • x = 12 or (12, 0) Smyrna High School Geometry CP

  50. How Do We Graph a Linear Equation??? Let’s Practice: 20x + 10y = 240 • Find the y-intercept: Let x = 0 and solve the equation for y. • 20(0) + 10y= 240 • 10x = 240 • 10 10 • y = 24 or (0, 24) Smyrna High School Geometry CP