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Solving Linear Equations

Solving Linear Equations. Objective: I will Use properties of numbers to solve linear equations with rational number coefficients. –64 8. Warm Up Add, subtract, multiply, or divide. 2. 23 – 19. 1. 24 + 17. 41. 4. 3. 12  3. 36. 4. 6(–7). –42. 5. –8. 6. –250 + (–85).

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Solving Linear Equations

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  1. Solving Linear Equations Objective: I will Use properties of numbers to solve linear equations with rational number coefficients

  2. –64 8 Warm Up Add, subtract, multiply, or divide. 2. 23 – 19 1. 24 + 17 41 4 3. 12  3 36 4. 6(–7) –42 5. –8 6. –250 + (–85) –335

  3. Equation • mathematical sentence that uses an equal sign to show that two expressions have the same value Addition Property of Equality • Add the same number to each side of the equals sign If a = b, then a + c = b + c x – 5 = 2

  4. Subtraction Property of Equality • Subtract the same number to each side of the equals sign. If a = b, then a – c = b – c x + 2 = 7

  5. Inverse Operations Addition and subtraction are inverseoperations, which means they “undo” each other. * To solve an equation, use inverse operations to isolate the variable. In other words, get the variable alone on one side of the equal sign.

  6. ? x + 8 = 15 ? 5 + 8 = 15 ? 13= 15 Example 1: Determining Whether a Number is a Solution of an Equation Determine which value of x is a solution of the equation. x + 8 = 15; x = 5, 7, or 23 Substitute each value for x in the equation. Substitute 5 for x.  So 5 is not solution.

  7. ? x + 8 = 15 ? 7 + 8 = 15 ? 15= 15 Example 1 Continued Determine which value of x is a solution of the equation. x + 8 = 15; x = 5, 7, or 23 Substitute each value for x in the equation. Substitute 7 for x.  So 7 is a solution.

  8. ? x + 8 = 15 ? 23 + 8 = 15 ? 31= 15 Example 1 Continued Determine which value of x is a solution of the equation. x + 8 = 15; x = 5, 7, or 23 Substitute each value for x in the equation. Substitute 23 for x.  So 23 is not a solution.

  9. ? 10 + 8 = 18 ? 18 = 18 Solving Equations Using Addition and Subtraction Properties Solve. 10 + n = 18 Use the Subtraction Property of Equality: Subtract 10 from both sides. 10 + n = 18 –10 –10 0 + n = 8 Identity Property of Zero: 0 + n = n. n = 8 Check 10 + n = 18 Substitute 8 for n. 

  10. ? 17 – 8 = 9 ? 9 = 9 Solving Equations Using Addition and Subtraction Properties Solve. p – 8 = 9 p – 8 = 9 Use the Addition Property of Equality: Add 8 to both sides. + 8 + 8 p + 0= 17 Identity Property of Zero: p + 0 = p. p = 17 Check p – 8 = 9 Substitute 17 for p. 

  11. ? 22 = 33 – 11 ? 22 = 22 Solving Equations Using Addition and Subtraction Properties Solve. 22 = y – 11 22 = y – 11 Use the Addition Property of Equality: Add 11 to both sides. + 11 + 11 33 = y + 0 Identity Property of Zero: y + 0 = y. 33 = y Check 22 = y – 11 Substitute 33 for y. 

  12. Helpful Hint! Force is measured in newtons (N). The number of newtons tells the size of the force and the sign tells the direction. Positive is to the right, and negative is to the left. Problem Solving Application Jan and Alex are arguing over who gets to play a board game. If Jan, on the right, pulls with a force of 14 N, what force is Alex exerting on the game if the net force is 3 N?

  13. 1 Understand the Problem Jan’s force Net force Alex’s force = + The answer is the force that Alex, on the left, is exerting on the board game. List the important information: • Jan, on the right pulls with a force of 14 N. • The net force is 3 N. Show the relationship or the information:

  14. 2 Make a Plan 3 Solve – 14 – 14 Write an equation and solve it. Let f represent Alex’s force on the board game, and use the equation model. 3 = f + 14 3 = f + 14 Subtract 14 from both sides. –11 = f Alex was exerting a force of –11 N on the board game.

  15. 4 Look Back Check the answer by using a number line. Move 14 units right to show Jan's force. Move 11 units to the left to show Alex's force. 11 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

  16. Multiplication Property of Equality • Multiply each side by the same nonzero number. If a = b and c ≠ 0, then a ∙ c = b ∙ c

  17. Division Property of Equality • Divide each side by the same nonzero number. If a = b, and c ≠ 0, then a ÷ c = b ÷ c 5x = 30

  18. Solve.

  19. Solve.

  20. More Practice

  21. Solve.

  22. Solve.

  23. Solve.

  24. Solve.

  25. Solve these multi-step equations.

  26. Solve.

  27. Example 2 During one shift, a waiter earns wages of $30 and gets an additional 15% in tips on customers’ food bills. The waiter earns $105. What is the total of the customers’ food bills?

  28. Guided Practice A real estate agent’s base salary is $22,000 per year. The agent earns a 4% commission on total sales. How much must the agent sell to earn $60,000 in one year?

  29. Example 5 It takes you 8 minutes to wash a car and it takes a friend 6 minutes to wash a car. How long does it take the two of you to wash 7 cars if you work together?

  30. Homework

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