1. Section 2.4 Solving Linear Equations in One Variable Using the Addition-Subtraction Principle

2. 2.4 Lecture Guide: Solving Linear Equations in One Variable Using the Addition-Subtraction Principle Objective: Solve linear equations in one variable using the addition-subtraction principle.

3. Linear Equation in One Variable Algebraically A linear equation in one variable x is an equation that can be written in the form, where A and B are real constants and. Verbally A linear equation in one variable is first degree in this variable. Algebraic Example

4. 1. Which of the following choices are linear equations in one variable? (a) (b) (c) (d)

5. Addition-Subtraction Principle of Equality Verbally If the same number is added to or subtracted from both sides of an equation, the result is an equivalent equation. Algebraically If a, b, and c are real numbers, then is equivalent to and to Numerical Example is equivalent to and to

6. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution of each equation. 2.

7. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 3.

8. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 4.

9. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 5.

10. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 6.

11. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 7.

12. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 8.

13. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 9.

14. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 10.

15. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 11.

16. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 12.

17. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 13.

18. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 14.

19. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 15.

20. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 16.

21. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 17.

22. Use the addition-subtraction principle of equality to solve each equation. Note that we can check our solution to each equation. 18.

23. Based on the limited variety of equations we have examined, a good strategy to solve a linear equation in one variable is: 1. Remove any ____________ symbols. 2. Use the addition-subtraction principle of equality to move all ____________ terms to one side. 3. Use the addition-subtraction principle of equality to move all ____________ terms to the other side.

24. Objective: Identify a linear equation as a conditional equation, and identity, or a contradiction. There are three classifications of linear equations to be aware of: conditional equations, identities, and contradictions. Each of the equations in problems 2-18 is called a __________________ __________________ because it is only true for certain values of the variable and untrue for other values. The following table compares all three types of linear equations.

25. Conditional Equation, Identity, and a Contradiction Conditional Equation Verbally A conditional equation is true for some values of the variable and false for other values. Algebraic Example Answer: The only value of x that checks is. Numerical Example

26. Conditional Equation, Identity, and a Contradiction Identity Verbally An identity is an equation that is true for all values of the variable. Algebraic Example Answer: All real numbers. Numerical Example All real numbers will check. is always.

27. Conditional Equation, Identity, and a Contradiction Contradiction Verbally A contradiction is an equation that is false for all values of the variable. Algebraic ExampleNumerical Example Answer: No solution. No real numbers will check because no real number is 3 greater than its own value.

28. If the solution process for solving a linear equation in one variable produces a unique solution, then the original equation is a conditional equation. If the solution process results in the variable disappearing from both sides of the equation, then the equation you are trying to solve is either a contradiction or an identity

29. Identify each contradiction or identity and write the solution of the equation. 19.

30. Identify each contradiction or identity and write the solution of the equation. 20.

31. Identify each contradiction or identity and write the solution of the equation. 21.

32. Identify each contradiction or identity and write the solution of the equation. 22.

33. Simplify vs Solve Simplify the expression in the first column by combining like terms, and solve the equation in the second column. 23. 24.

34. Simplify vs Solve Simplify the expression in the first column by combining like terms, and solve the equation in the second column. 25. 26.

35. Objective: Use tables and graphs to solve a linear equation in one variable. The solution of a linear equation in one variable is an x- value that causes both sides of the equation to have the same value. To solve a linear equation in one variable using tables or graphs, let equal the left side of the equation and letequal the right side of the equation. Using a table of values, look for the ______-value where the two ______-values are equal. Using a graph, look for the ______-coordinate of the point of __________________ of the two graphs. Note that the solution of a linear equation in one variable is an x-value and not an ordered pair.

36. 27. Use the table shown to determine the solution of the equation. The x-value in the table at which the two y values are equal is ______. Solution: _____________ Verify your result by solvingalgebraically.

37. 28. Use the graph shown to determine the solution of the equation. The point where the two lines intersect has an x-coordinate of ______. Solution: _____________ Verify your result by solvingalgebraically.

38. 29. Solve each equation using a table or a graph from your calculator by letting equal the left side of the equation and equal the right side of the equation. See Calculator Perspective 2.4.1 for help. Solution: ____________

39. 30. Solve each equation using a table or a graph from your calculator by letting equal the left side of the equation and equal the right side of the equation. See Calculator Perspective 2.4.1 for help. Solution: ____________

40. Once the viewing window has been adjusted so you can see the point of intersection of two lines, the keystrokes required to find that point of intersection are ______ ______ ______ ______ ______ ______. To view a graph in the standard viewing window, press ZOOM ______.

41. 31. Examine what happens if we try solving by entering the left side as and the right side ason a graphing calculator. (a) Do the two graphs appear to intersect?

42. 31. Examine what happens if we try solving by entering the left side as and the right side ason a graphing calculator. (b) Compare the values of and for each x-value in the table. What do you observe?

43. 31. Examine what happens if we try solving by entering the left side as and the right side ason a graphing calculator. (c) Now solve algebraically. Is this equation a conditional equation, a contradiction, or an identity?

44. Translate each verbal statement into algebraic form. 32. Three times a number z.

45. Translate each verbal statement into algebraic form. 33. Three less than a number w.

46. Translate each verbal statement into algebraic form. 34. Seven less than six times a number a.

47. Translate each verbal statement into algebraic form. 35. Two times the quantity of eight less than a number x.

48. 36. Write an algebraic equation for the following statement, using the variable m to represent the number, and then solve for m. Verbal Statement: Five less than three times a number is equal to two times the sum of the number and three. Algebraic Equation: Solve this equation:

49. 37. The perimeter of the parallelogram shown equals. Find a. a a 8 8