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

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

1. Mastering Linear Equations in One Variable

2. What are Linear Equations? Equations with one variable (usually x) The variable has an exponent of 1 Example: 2x + 5 = 13 Used to find an unknown value Can you think of real-life situations where we might use linear equations?

3. Parts of a Linear Equation Variable: The unknown value (usually x) Coefficients: Numbers multiplied by the variable Constants: Numbers not attached to the variable Equal sign: Shows the two sides are balanced What are the parts in this equation: 3x - 7 = 11?

4. Goal: Isolate the Variable We want to get the variable (x) by itself Whatever we do to one side, we must do to the other This keeps the equation balanced Think of it like a scale - both sides must always be equal

5. Step 1: Combine Like Terms Group similar terms on each side of the equation Add or subtract constants Add or subtract terms with variables Example: 2x + 5 + 3x = 20 + 7 Becomes: 5x + 5 = 27 Can you combine like terms in 4x + 3 - 2x = 15?

6. Step 2: Move Variables to One Side Choose a side for the variable (usually the left) Add or subtract to move variable terms Remember: Do the same to both sides! Example: x + 7 = 15 Subtract 7 from both sides: x = 8

7. Step 3: Move Constants to the Other Side Move constant terms to the opposite side of the variable Add or subtract as needed Example: 2x - 5 = 11 Add 5 to both sides: 2x = 16 What would you do to solve 3x + 8 = 23?

8. Step 4: Divide or Multiply If the variable has a coefficient, divide both sides by it This isolates the variable (x) Example: 2x = 16 Divide both sides by 2: x = 8 Always check: Is your variable (x) alone?

9. Practice: Solve Step-by-Step Let's solve: 3x + 4 = 19 Step 1: 3x + 4 = 19 (No like terms to combine) Step 2: 3x = 15 (Subtract 4 from both sides) Step 3: x = 5 (Divide both sides by 3) Can you explain each step to a classmate?

10. Equations with Variables on Both Sides Sometimes variables appear on both sides Example: 2x + 3 = 5x - 9 Goal: Get all variables on one side Subtract 2x from both sides: 3 = 3x - 9 Now solve as usual Why do you think we chose to subtract 2x instead of 5x?

11. Solving Equations with Fractions Multiply both sides by the denominator to eliminate fractions Example: (1/3)x + 2 = 8 Multiply everything by 3: x + 6 = 24 Now solve as usual How would you solve (1/2)x - 3 = 7?

12. Real-World Application: Age Problems "I'm thinking of a number. Twice my number plus 7 equals 25. What's my number?" Let x be the number: 2x + 7 = 25 Solve step-by-step Solution: x = 9 Can you create a similar problem for your partner to solve?

13. Real-World Application: Money Problems "You have $12 more than twice your savings. You have $60 in total. How much did you save?" Let x be savings: 2x + 12 = 60 Solve it! Solution: x = $24 Why are linear equations useful for solving money problems?

14. Common Mistakes to Avoid Forgetting to do the same operation on both sides Dividing only the coefficient (not the whole side) by a number Combining unlike terms Can you think of other mistakes to watch out for?

15. Check Your Solution Always substitute your answer back into the original equation Example: If 2x + 5 = 13 and you found x = 4, check: 2(4) + 5 = 13 8 + 5 = 13 13 = 13 (It works!) Why is checking your solution important?

16. Creating Your Own Equations Start with a number in mind (the solution) Build an equation around it Example: Solution is 5 2x + 3 = 13 (because 2(5) + 3 = 13) Try creating an equation for your partner to solve!

17. Equation or Expression? Equation: Has an equal sign (2x + 3 = 11) Expression: No equal sign (2x + 3) We solve equations, but simplify expressions Can you give an example of each?

18. Word Problem Strategy Read the problem carefully Identify the unknown (your variable) Write an equation using the information given Solve the equation Check if your answer makes sense What's the hardest part of word problems for you?

19. Practice Makes Perfect! Solving linear equations takes practice Start with simple equations and work your way up Use online resources for extra practice Don't be afraid to ask for help Remember: Every expert was once a beginner!

20. Review and Questions What are the key steps to solving a linear equation? Why is it important to do the same thing to both sides? What was the most challenging concept for you? Any questions about what we've covered?

21. Introduction to Linear Equations Linear equations have one variable (usually x) The variable always has an exponent of 1 Used to find unknown values Example: 2x + 5 = 13 Can you think of a real-life situation where you might use a linear equation?

22. Parts of a Linear Equation Variable: The unknown value (usually x) Coefficient: Number multiplied by the variable Constant: Number not attached to the variable Equal sign: Shows the two sides are balanced Can you identify these parts in the equation 3x - 7 = 11?

23. Goal: Isolate the Variable We want to get the variable (x) by itself Whatever we do to one side, we must do to the other This keeps the equation balanced Think of it like a scale - both sides must always be equal Why do you think it's important to keep the equation balanced?

24. Step 1: Combine Like Terms Group similar terms on each side of the equation Add or subtract constants Add or subtract terms with variables Example: 2x + 5 + 3x = 20 + 7 becomes 5x + 5 = 27 Try combining like terms in this equation: 4x + 3 - 2x = 15

25. Step 2: Move Variables to One Side Choose a side for the variable (usually the left) Add or subtract to move variable terms Remember: Do the same to both sides! Example: x + 7 = 15, subtract 7 from both sides: x = 8 How would you move the variable in this equation: 3 = x - 5?

26. Step 3: Move Constants to the Other Side Move constant terms to the opposite side of the variable Add or subtract as needed Example: 2x - 5 = 11, add 5 to both sides: 2x = 16 What would you do to solve 3x + 8 = 23?

27. Step 4: Divide or Multiply If the variable has a coefficient, divide both sides by it This isolates the variable (x) Example: 2x = 16, divide both sides by 2: x = 8 Always check: Is your variable (x) alone? Why is this step important in solving linear equations?

28. Practice: Solve Step-by-Step Let's solve: 3x + 4 = 19 Step 1: 3x + 4 = 19 (No like terms to combine) Step 2: 3x = 15 (Subtract 4 from both sides) Step 3: x = 5 (Divide both sides by 3) Can you explain each step to a classmate?

29. Equations with Variables on Both Sides Sometimes variables appear on both sides Example: 2x + 3 = 5x - 9 Goal: Get all variables on one side Subtract 2x from both sides: 3 = 3x - 9 Now solve as usual Why do you think we chose to subtract 2x instead of 5x?

30. Solving Equations with Fractions Multiply both sides by the denominator to eliminate fractions Example: (1/3)x + 2 = 8 Multiply everything by 3: x + 6 = 24 Now solve as usual How would you solve (1/2)x - 3 = 7?

31. Real-World Application: Age Problems "I'm thinking of a number. Twice my number plus 7 equals 25. What's my number?" Let x be the number: 2x + 7 = 25 Solve step-by-step Solution: x = 9 Can you create a similar problem for your partner to solve?

32. Real-World Application: Money Problems "You have $12 more than twice your savings. You have $60 in total. How much did you save?" Let x be savings: 2x + 12 = 60 Solve it! Solution: x = $24 Why are linear equations useful for solving money problems?

33. Common Mistakes to Avoid Forgetting to do the same operation on both sides Dividing only the coefficient (not the whole side) by a number Combining unlike terms Can you think of other mistakes to watch out for? How can you prevent these mistakes in your own work?

34. Check Your Solution Always substitute your answer back into the original equation Example: If 2x + 5 = 13 and you found x = 4, check: 2(4) + 5 = 13 8 + 5 = 13 13 = 13 (It works!) Why is checking your solution important?

35. Creating Your Own Equations Start with a number in mind (the solution) Build an equation around it Example: Solution is 5 2x + 3 = 13 (because 2(5) + 3 = 13) Try creating an equation for your partner to solve! How does creating equations help you understand them better?

36. Equation or Expression? Equation: Has an equal sign (2x + 3 = 11) Expression: No equal sign (2x + 3) We solve equations, but simplify expressions Can you give an example of each? Why is it important to know the difference?

37. Word Problem Strategy Read the problem carefully Identify the unknown (your variable) Write an equation using the information given Solve the equation Check if your answer makes sense What's the hardest part of word problems for you?

38. Practice Makes Perfect! Solving linear equations takes practice Start with simple equations and work your way up Use online resources for extra practice Don't be afraid to ask for help Remember: Every expert was once a beginner! What's your favorite way to practice math?

39. Review and Questions What are the key steps to solving a linear equation? Why is it important to do the same thing to both sides? What was the most challenging concept for you? Any questions about what we've covered? How can you apply what you've learned to other areas of math?

40. 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?

41. Question 2 Which of the following is NOT a linear equation in one variable? A) 2x + 7 = 15 B) y = 3x + 2 C) 4x - 9 = x + 6 D) x^2 + 3 = 11 Why did you choose your answer?

42. Question 3 What is the first step in solving 2(x + 3) = 14? A) Subtract 3 from both sides B) Divide both sides by 2 C) Distribute the 2 D) Add 3 to both sides How does this step help simplify the equation?

43. Question 4 Solve: 1/2x - 4 = 6 A) x = 20 B) x = 5 C) x = 10 D) x = 15 What did you do to eliminate the fraction in this equation?

44. Question 5 Tom is 5 years older than twice Sarah's age. If Tom is 31, how old is Sarah? A) 13 B) 18 C) 15 D) 11 How did you translate this word problem into an equation?

45. Question 6 Which step is incorrect in solving 3x - 7 = 14? 1. 3x = 21 2. x = 21/3 3. x = 7 A) Step 1 B) Step 2 C) Step 3 D) All steps are correct Explain why the step you chose is incorrect or why all steps are correct.

46. Question 7 Simplify: 5x + 3 - 2x + 7 = 20 A) 3x + 10 = 20 B) 7x + 10 = 20 C) 3x - 4 = 20 D) 5x + 10 = 20 What strategy did you use to combine like terms?

47. Question 8 Solve: 4(x - 2) = 24 A) x = 8 B) x = 6 C) x = 10 D) x = 4 What property did you use to solve this equation?

48. Question 9 You have $15 more than three times your savings. If you have $60 in total, how much did you save? A) $20 B) $15 C) $30 D) $45 How did you set up the equation for this problem?

49. Question 10 Which equation is equivalent to 2x + 5 = 3x - 1? A) -x = -6 B) x = 6 C) x = -6 D) -x = 6 What steps did you take to get all variables on one side of the equation?

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