Linear Equations

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Use this PowerPoint to review Linear Equations in preparation to your Unit 3 Test.

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Linear Equations

  1. 1. Linear Equations and Inequalities
  2. 2. Properties of Equality If a = b, then a + c = b + c. Multiplication Property of Equality If a = b, then a – c = b – c. Addition Property of Equality If a = b, then a × c = b × c. Division Property of Equality If a = b, then a ÷ c = b ÷ c. Subtraction Property of Equality Let a, b, and c be real numbers.
  3. 3. Properties of Operations a + b = b + a Distributive Property (a + b) + c = a + (b + c) Commutative Property a × (b + c) = (a × b) + (a × c) Associative Property a × b = b × a (a × b) × c = a × (b × c) a × (b – c) = (a × b) – (a × c)
  4. 4. Properties of Inequality Multiplication Property of Inequality Addition Property of Inequality Division Property of Inequality Subtraction Property of Inequality If a > b, c > 0, then a ÷ c > b ÷ c. If a > b, c < 0, then a ÷ c < b ÷ c. If a < b, c < 0, then a ÷ c > b ÷ c. If a < b, c > 0, then a ÷ c < b ÷ c. If a > b, c > 0, then a × c > b × c, If a > b, c < 0, then a × c < b × c. If a < b, c < 0, then a × c > b × c. If a < b, c > 0, then a × c < b × c. What happens when you replace < with > or < or > in each of these properties? Does the property still hold? Try it! Let a, b, and c be real numbers. If a > b, then a + c > b + c. If a < b, then a + c < b + c. If a > b, then a – c > b – c. If a < b, then a – c < b – c.
  5. 5. Linear Equations with One Variable An equation can be either a true statement or a false statement. Which of these equations are true statements? 3(4 + 5) = 27 12 – 3 = 3 – 12 8 + 2 = 9 9 + 1 = 10 6 × 5 = 5 × 6 42 = 8 An equation with a variable can be either a true statement or a false statement, depending on the value given to the variable. Which of these equations are true statements? 3(h + 5) = 27, h = 48 + x = 9, x = 1 6 × 5 = r × 6, r = 4 12 – 3 = 3 – m, m = –6y + 1 = 10, y = 8 42 = w, w = 8
  6. 6. Linear Equations with One Variable An equation can be either a true statement or a false statement. Which of these equations are true statements? 3(4 + 5) = 27 12 – 3 = 3 – 12 8 + 2 = 9 9 + 1 = 10 6 × 5 = 5 × 6 42 = 8 An equation with a variable can be either a true statement or a false statement, depending on the value given to the variable. Which of these equations are true statements? 3(h + 5) = 27, h = 48 + x = 9, x = 1 6 × 5 = r × 6, r = 4 12 – 3 = 3 – m, m = –6y + 1 = 10, y = 8 42 = w, w = 8
  7. 7. Solving Linear Equations with One Variable When solving an equation, you find the value(s) of the variable that makes the equation true. Use your number sense to help you solve an equation with one variable. y + 2 = 11 think What number plus 2 equals 11? next Substitute the number you have chosen into the original equation to check to see that it makes the equation true. y + 2 = 11 9 + 2 = 11 11 = 11 Let y = 9. Substitute 9 for y. The equation is true for y = 9.
  8. 8. Solving Linear Equations with One Variable Use properties to help you solve an equation with one variable. y + 2 = 11 y + 2 – 2 = 11 – 2 y = 9 The solution checks. Using the Subtraction Property of Equality, subtract 2 from both sides of the equation. Check the solution. y + 2 = 11 9 + 2 = 11 11 = 11 Check:
  9. 9. Solving Linear Inequalities with One Variable x – 4 > 2 x – 4 + 4 > 2 + 4 x > 6 Using the Addition Property of Equality, add 4 to both sides of the equation. Graph the solution on a number line. x – 4 > 2 Use Properties to help you solve an inequality with one variable. 5 9-3 1 6 10-2 2-4 -1 0 3 4 7 8 x – 4 > 2 7 – 4 > 2 3 > 2 All values greater than 6 are solutions to this inequality. The solution checks. Try x = 7. Choose a value greater than 6 and check it.
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