1. 2-1 Solving One-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 2-1 Solving One-Step Equations Warm Up Evaluate. 1. – + 4 3. 0.96 ÷ 6 Evaluate each expression for a = 3 and b = –2. 6. a + 57. 12b –0.8 1 3 2 3 3 2 3 –24 8 2. –0.51 + (–0.29) 4. (–9)(–9) 0.16 81 5. 1

3. 2-1 Solving One-Step Equations Preparation for 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable, and provide justification for each step. Also covered: 2.0 California Standards

4. 2-1 Solving One-Step Equations equation solution of an equation solution set Vocabulary

5. 2-1 Solving One-Step Equations An equation is a mathematical statement that two expressions are equal. A solution of an equation is a value of the variable that makes the equation true. A solution set is the set of all solutions. Finding the solutions of an equation is also called solving the equation.

6. 2-1 Solving One-Step Equations Inverse Operations Add x. Subtract x. Multiply by x. Divide by x. An equation is like a balanced scale. To keep the balance, you must perform the same inverse operation on both sides. To find solutions, perform inverse operations until you have isolated the variable. A variable is isolated when it appears by itself on one side of an equation, and not at all on the other side.

7. 2-1 Solving One-Step Equations

8. 2-1 Solving One-Step Equations Solution sets are written in set notation using braces, { }. Solutions may be given in set notation, or they may be given in the form x = 14. Writing Math

9. 2-1 Solving One-Step Equations Solve the equation. Additional Example 1A: Solving Equations by Using Addition or Subtraction Since 8 is subtracted from y, add 8 to both sides to undo the subtraction. y – 8 = 24 + 8 + 8 y = 32 Check y – 8 = 24 32 – 8 24 24 To check your solution, substitute 32 for y in the original equation. The solution set is {32}.

10. 2-1 Solving One-Step Equations Solve the equation. Additional Example 1B: Solving Equations by Using Addition To check your solution, substitute 2.4 for t in the original equation. Since 1.8 is added to t, subtract 1.8 from both sides to undo the addition. 4.2 = t + 1.8 –1.8 –1.8 2.4 = t Check 4.2 = t + 1.8 4.2 2.4 + 1.8 4.2 The solution set is {2.4}.

11. 2-1 Solving One-Step Equations Solve the equation. Check your answer. Check It Out! Example 1a Since 3.2 is subtracted from n, add 3.2 to both sides to undo the subtraction. n – 3.2 = 5.6 + 3.2 + 3.2 n = 8.8 Checkn – 3.2 = 5.6 8.8 – 3.2 5.6 5.6 To check your solution, substitute 8.8 for n in the original equation. The solution set is {8.8}.

12. 2-1 Solving One-Step Equations Solve the equation. Check your answer. Check It Out! Example 1b Since 6 is subtracted from k, add 6 to both sides to undo the subtraction. –6 = k – 6 + 6 + 6 0 = k Check –6 = k – 6 –6 0 – 6 –6 To check your solution, substitute 0 for k in the original equation. The solution set is {0}.

13. 2-1 Solving One-Step Equations Solve the equation. Check your answer. Check It Out! Example 1c Since 6 is added to t, subtract 6 from both sides to undo the addition. 6 + t = 14 – 6 – 6 t = 8 Check 6 + t = 14 6 + 8 14 14 To check your solution, substitute 8 for t in the original equation. The solution set is {8}.

14. 2-1 Solving One-Step Equations

15. 2-1 Solving One-Step Equations Solve the equation. Check your answer. Additional Example 2A: Solving Equations by Using Multiplication or Division Since j is divided by 3, multiply from both sides by 3 to undo the division. –8 –8 To check your solution, substitute –24 for j in the original equation. –24 = j Check The solution set is {–24}.

16. 2-1 Solving One-Step Equations Solve the equation. Check your answer. Additional Example 2B: Solving Equations by Using Multiplication or Division Since v is multiplied by –6, divide both sides by –6 to undo the multiplication. –4.8 = –6v 0.8 = v Check –4.8 = –6v –4.8 –6(0.8) –4.8 To check your solution, substitute 0.8 for v in the original equation. The solution set is {0.8}.

17. 2-1 Solving One-Step Equations Solve each equation. Check your answer. Check It Out! Example 2a Since p is divided by 5, multiply both sides by 5 to undo the division. To check your solution, substitute 50 for p in the original equation. p = 50 Check 10 The solution set is {50}.

18. 2-1 Solving One-Step Equations Solve each equation. Check your answer. Check It Out! Example 2b Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication. 0.5y = –10 Check 0.5y = –10 0.5(–20) –10 –10 To check your solution, substitute –20 for y in the original equation. y = –20 The solution set is {–20}.

19. 2-1 Solving One-Step Equations Solve each equation. Check your answer. Check It Out! Example 2c Since c is divided by 8, multiply both sides by 8 to undo the division. c = 56 To check your solution, substitute 56 for c in the original equation. 7 The solution set is {56}. Check

20. 2-1 Solving One-Step Equations When solving equations, you will sometimes find it easier to add an opposite to both sides instead of subtracting or to multiply by a reciprocal instead of dividing. This is often true when an equation contains negative numbers or fractions.

21. 2-1 Solving One-Step Equations Additional Example 3A: Solving Equations by Using Opposites or Reciprocals Solve each equation. The reciprocal of is. Since w is multiplied by multiply both sides by. The solution set is {–24}.

22. 2-1 Solving One-Step Equations Additional Example 3B: Solving Equations by Using Opposites or Reciprocals Solve each equation. Since p is added to, add to both sides to undo the subtraction. The solution set is. { }

23. 2-1 Solving One-Step Equations Solve the equation. Check your answer. Check It Out! Example 3a Since –2.3 is added to m, add 2.3 to both sides. –2.3 + m = 7 +2.3 + 2.3 m = 9.3 –2.3 + m = 7 The solution set is {9.3}. To check your solution, substitute 9.3 for m in the original equation. Check –2.3 + m = 7 –2.3 + 9.3 7 7

24. 2-1 Solving One-Step Equations 5 4 + z = Since is added to z add to both sides. The solution set is {2}. Solve the equation. Check your answer. Check It Out! Example 3b To check your solution, substitute 2 for z in the original equation. Check

25. 2-1 Solving One-Step Equations Check It Out! Example 3c w = 612 Solve the equation. Check your answer. The reciprocal of is. Since w is multiplied by multiply both sides by. Check 102 To check your solution, substitute 612 for w in the original equation. The solution set is {612}.

26. 2-1 Solving One-Step Equations Additional Example 4: Application Ciro deposits of the money he earns from mowing lawns into a college education fund. This year Ciro added $285 to his college education fund. Write and solve an equation to find out how much money Ciro earned mowing lawns this year. 1 4

27. 2-1 Solving One-Step Equations Additional Example 4 Continued e = $1140 The original earnings were $1140. Write an equation to represent the relationship. earningsistimes$285 1 4 1 4 4 1  e = 285 4 1  The reciprocal of is. Since e is multiplied by, multiply both sides by 1 4 1 4 4 1 4 1.  e = $285

28. 2-1 Solving One-Step Equations Check It Out! Example 4 The distance in miles from the airport that a plane should begin descending divided by 3 equals the plane’s height above the ground in thousands of feet. A plane is 10,000 feet above the ground. Write and solve an equation to find the distance from the airport at which this plane should begin descending.

29. 2-1 Solving One-Step Equations Check It Out! Example 4 Continued d ÷ 3 = h Write an equation to represent the relationship. d = 30 At 10,000 feet altitude the decent should start 30,000 feet from the airport. distance divided by height is 3 3 1  d 3 = 10 3 1  Substitute 10 for h. The reciprocal of is. Since d is multiplied by multiply both sides by. 1 3 3 1 1 3 3 1

30. 2-1 Solving One-Step Equations Lesson Quiz Solve each equation. 1. r – 4 = –8 3. 5. 6. This year a high school had 578 sophomores enrolled. This is 89 less than the number enrolled last year. Write and solve an equation to find the number of sophomores enrolled last year. –4 2.8 s – 89 = 578; s = 667 2. 4. 8y = 4 40