MATHEMATICS8 Q3 2. solve problems (e.g., number problems, geometry problems, and money problems).pptx

MATHEMATICS8 Q3 1. solve linear equations in one variable.pptx

1. Solving Linear Equations: From Numbers to Real-World Problems

2. Introduction to Linear Equations A linear equation is an equation where the variable has a power of 1 These equations form straight lines when graphed We'll learn to solve various types of linear equations You'll see how these apply to real-world situations

3. The Basics: Solving Number Problems Number problems involve finding an unknown value Example: 2x + 5 = 13 Can you guess what x might be? We'll learn step-by-step methods to solve these

4. Steps to Solve Number Problems Step 1: Isolate the variable on one side Step 2: Perform inverse operations Step 3: Check your answer Let's practice with: 3x - 7 = 14

5. Geometry Problems: Introduction Geometry problems often involve shapes and measurements We use linear equations to find unknown lengths or angles These problems help us understand real-world structures What geometric shapes do you see around you?

6. Example: Rectangle Perimeter Problem The perimeter of a rectangle is 24 cm Its length is 2 cm more than its width Can you write an equation to find the width? Let's solve it together!

7. Solving the Rectangle Problem Equation: 2(w + 2) + 2w = 24 Simplify: 2w + 4 + 2w = 24 Combine like terms: 4w + 4 = 24 Solve for w. What did you get?

8. Money Problems: Real-World Applications Money problems connect math to everyday life These often involve budgeting, saving, or spending Linear equations help us make financial decisions What money-related decisions have you made recently?

9. Example: Savings Goal Problem You want to save $500 for a new bike You already have $75 saved You can save $25 per week How many weeks will it take to reach your goal?

10. Solving the Savings Problem Let w be the number of weeks Equation: 75 + 25w = 500 Subtract 75 from both sides: 25w = 425 Divide both sides by 25 How many weeks did you calculate?

11. The Power of Variables Variables allow us to represent unknown quantities They help us create general solutions This makes problem-solving more efficient Can you think of other situations where variables are useful?

12. Translating Words to Equations Key to solving word problems is translating text to math "is" often means equals (=) "more than" usually indicates addition (+) "less than" typically suggests subtraction (-) Practice: "Five more than twice a number is 17"

13. Checking Your Solutions Always verify your answer by substituting it back into the original equation This helps catch calculation errors It builds confidence in your problem-solving skills How might you check the solution to 3x + 7 = 22?

14. Real-World Application: Distance Problems Distance = Rate × Time If you know two variables, you can find the third Example: How long to travel 240 miles at 60 mph? Can you write and solve the equation?

15. Solving Equations with Fractions Sometimes, our equations involve fractions Key strategy: Multiply both sides by the denominator This eliminates fractions, making the equation easier Try solving: 1/3x + 2 = 8

16. Equations in Science Linear equations are used in many scientific fields In physics: Force = Mass × Acceleration In chemistry: Density = Mass / Volume How might these equations help scientists solve problems?

17. Creating Your Own Problems Now it's your turn to be creative! Think of a real-life situation that could involve a linear equation Write a problem and its corresponding equation Trade with a classmate and solve each other's problems

18. Group Challenge: The Lemonade Stand You're running a lemonade stand Lemonade costs $0.50 per cup to make You sell it for $1.50 per cup How many cups must you sell to make $20 profit? Work in groups to solve this problem

19. Reflection and Review What was the most interesting problem you solved today? Which type of problem did you find most challenging? How might you use linear equations in your daily life? What questions do you still have about solving linear equations?

20. Next Steps: Graphing Linear Equations We've solved linear equations algebraically Next, we'll learn to graph these equations This will give us a visual representation of solutions How do you think graphing might help in problem-solving?

21. Conclusion: The Power of Linear Equations Linear equations help us solve various real-world problems They connect different areas: numbers, geometry, money, science Mastering these skills opens doors to advanced math concepts Remember: Practice makes perfect. Keep solving problems!

22. What are Linear Equations? Equations where the variable has a power of 1 Form straight lines when graphed Examples: y = 2x + 3, 3x - 7 = 14 Can you think of any real-life situations that might use linear equations?

23. Components of a Linear Equation Variables: Unknown quantities (usually x or y) Coefficients: Numbers multiplied by variables Constants: Numbers without variables Equality sign: Shows the two sides are equal Can you identify these in the equation 2x + 5 = 13?

24. Solving One-Step Equations Simplest form of linear equations Involve only one operation to solve Examples: x + 5 = 12, 3y = 21 How would you solve x - 8 = 7?

25. Two-Step Equations Require two operations to solve Often involve both addition/subtraction and multiplication/division Example: 2x + 3 = 11 Can you solve 3y - 4 = 14?

26. Equations with Variables on Both Sides More complex equations Variables appear on both sides of the equal sign Example: 3x + 2 = x + 10 Strategy: Move all variables to one side What's your first step in solving 5x - 3 = 2x + 9?

27. Solving Equations with Fractions Equations that involve fractions Key strategy: Multiply both sides by the common denominator Example: (1/3)x + 2 = 8 How would you start solving (1/4)x - 2 = 3?

28. Word Problems: Translating Words to Equations Key skill in problem-solving Common phrases and their mathematical meanings: "is" often means equals (=) "more than" usually indicates addition (+) "less than" typically suggests subtraction (-) Can you translate "Five less than twice a number is 11" into an equation?

29. Age Problems Common type of word problem Often involve comparing ages of two people Example: "Mary is 5 years older than twice Tom's age. Mary is 35. How old is Tom?" How would you set up this equation?

30. Distance-Rate-Time Problems Based on the formula: Distance = Rate × Time Useful for travel and motion problems Example: "How long will it take to travel 240 miles at 60 mph?" Can you write the equation for this problem?

31. Mixture Problems Involve combining two or more substances Often used in chemistry or cooking Example: "How much 20% salt solution should be added to 5 liters of 50% salt solution to make a 30% salt solution?" What information do you need to solve this type of problem?

32. Work Problems Involve rates at which tasks are completed Often include multiple people working together Example: "Alice can paint a house in 6 hours. Bob can paint it in 8 hours. How long will it take them working together?" How would you approach solving this problem?

33. Investment Problems Involve calculating interest, profit, or loss Use concepts of simple or compound interest Example: "How much should you invest at 5% annual interest to have $1000 after 2 years?" What information is crucial for solving investment problems?

34. Geometry Applications Use linear equations to solve for unknown lengths or angles Often involve formulas for area or perimeter Example: "The width of a rectangle is 3 cm less than its length. If the perimeter is 26 cm, find the dimensions." How would you set up this equation?

35. Systems of Linear Equations Two or more linear equations solved together Methods: substitution, elimination, graphing Example: Solve x + y = 5 and 2x - y = 1 What real-life situations might require solving systems of equations?

36. Graphing Linear Equations Visual representation of linear equations Use x-y coordinate system Slope-intercept form: y = mx + b m represents slope, b is y-intercept How does changing m or b affect the graph?

37. Applications in Science Physics: Force = Mass × Acceleration Chemistry: Concentration = Mass / Volume Biology: Population Growth Models Can you think of other scientific applications of linear equations?

38. Linear Equations in Economics Supply and demand curves Break-even analysis Cost and revenue functions How might a business use linear equations to make decisions?

39. Technology and Linear Equations Spreadsheet software for solving and graphing Online calculators and graphing tools Programming languages for complex calculations How can technology help you solve linear equations more efficiently?

40. Review and Practice Key concepts: Variables, coefficients, constants Solving methods: One-step, two-step, variables on both sides Word problems: Translate, set up equation, solve Real-world applications: Age, distance, mixture, work, investment What areas do you feel most confident in? Where do you need more practice?

41. Question 1 Solve for x: 3x + 5 = 20 A) 5 B) 15 C) 7 D) 10 What steps did you take to solve this equation?

42. Question 2 The length of a rectangle is 3 cm more than its width. If the perimeter is 26 cm, what is the width? A) 5 cm B) 7 cm C) 4 cm D) 6 cm How did you set up the equation for this problem?

43. Question 3 If you save $30 per week, how many weeks will it take to save $420? A) 12 weeks B) 14 weeks C) 16 weeks D) 18 weeks What linear equation represents this situation?

44. Question 4 A car travels at 60 mph. How long will it take to travel 240 miles? A) 3 hours B) 4 hours C) 5 hours D) 6 hours Which formula did you use to solve this problem?

45. Question 5 Translate into an equation: "Five less than twice a number is 11" A) 2x - 5 = 11 B) 2x + 5 = 11 C) x - 5 = 11 D) 2(x - 5) = 11 What words helped you identify the operations in this equation?

46. Question 6 Alice can paint a room in 4 hours. Bob can paint it in 6 hours. How long will it take them working together? A) 2 hours B) 2.4 hours C) 3 hours D) 5 hours How did you approach solving this work problem?

47. Question 7 Solve: 1/2x + 3 = 11 A) x = 14 B) x = 16 C) x = 18 D) x = 20 What strategy did you use to eliminate the fraction in this equation?

48. Question 8 Solve the system of equations: y = 2x + 1 and y = -x + 7 A) (2, 5) B) (3, 4) C) (1, 3) D) (2, 3) Which method did you use to solve this system: substitution, elimination, or graphing?

49. Question 9 If 3 pencils and 2 erasers cost $2.10, and 2 pencils and 3 erasers cost $1.95, what is the cost of 1 pencil? A) $0.30 B) $0.45 C) $0.50 D) $0.60 How did you set up the equations for this problem?

50. Question 10 A bike's original price was $200. After a 15% discount, what is the new price? A) $170 B) $160 C) $180 D) $185 What equation did you use to calculate the discounted price?

51. ANSWER KEYS 1.A 2.A 3.B 4.B 5.A 6.B 7.B 8.A 9.C 10.A