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Properties of Equality

Properties of Equality. Properties of Equality. Properties are rules that allow you to balance, manipulate, and solve equations. Addition Property of Equality. Adding the same number to both sides of an equation does not change the equality of the equation. If a = b, then a + c = b + c.

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Properties of Equality

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  1. Properties of Equality

  2. Properties of Equality • Properties are rules that allow you to balance, manipulate, and solve equations

  3. Addition Property of Equality • Adding the same number to both sides of an equation does not change the equality of the equation. • If a = b, then a + c = b + c. • Ex: x = y, so x + 2 = y + 2

  4. Subtraction Property of Equality • Subtracting the same number to both sides of an equation does not change the equality of the equation. • If a = b, then a – c = b – c. • Ex: x = y, so x – 4 = y – 4

  5. Multiplication Property of Equality • Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. • If a = b, then ac = bc. • Ex: x = y, so 3x = 3y

  6. Division Property of Equality

  7. Reflexive Property of Equality • A number is equal to itself. (Think mirror) • a = a • Ex: 4 = 4

  8. Symmetric Property of Equality • If numbers are equal, they will still be equal if the order is changed (reversed). • If a = b, then b = a. • Ex: x = 4, then 4 = x

  9. Transitive Property of Equality • If numbers are equal to the same number, then they are equal to each other. • If a = b and b = c, then a = c. • Ex: If x = 8 and y = 8, then x = y

  10. Commutative Property • Changing the order of addition or multiplication does not matter • Addition: a + b = b + a • Multiplication: a ∙ b = b ∙ a

  11. Associative Property • The change in grouping of three or more terms/factors does not change their sum or product. • Addition: a + (b + c) = (a + b) + c • Multiplication: a ∙ (b ∙ c) = (a ∙ b) ∙ c

  12. Additive Identity Property • The sum of any number and zero is always the original number • a + 0 = a • Ex: 4 + 0 = 4

  13. Multiplicative Identity Property • The product of any number and one is always the original number. • Multiplying by one does not change the original number. • a ∙ 1 = a • Ex: 2 ∙ 1 = 2

  14. Additive Inverse Property • The sum of a number and its inverse (or opposite) is equal to zero. • a + (-a) = 0 • Ex: 2 + (-2) = 0

  15. Multiplicative Inverse Property • The product of any number and its reciprocal is equal to 1.

  16. Multiplicative Property of Zero • The product of any number and zero is always zero. • a ∙ 0 = 0 • Ex: 298 ∙ 0 = 0

  17. Solve Systems of Equations by Graphing

  18. Solve Systems of Equations by Graphing • Make sure each equation is in slope-intercept form: y = mx + b. • Graph each equation on the same graph paper. • The point where the lines intersect is the solution. If they don’t intersect then there’s no solution. • Check your solution algebraically.

  19. 1. Graph to find the solution. • 1. Graph to find the solution.

  20. 2. Graph to find the solution. • Graph to find the solution.

  21. 3. Graph to find the solution.

  22. Types of Solutions

  23. So basically…. • If the lines have the same y-intercept b, and the same slope m, then the system is consistent-dependent. • If the lines have the same slope m, but different y-intercepts b, the system is inconsistent. • If the lines have different slopes m, the system is consistent-independent.

  24. Solve Systems of Equations by Elimination

  25. Steps for Elimination: • Arrange the equations with like terms in columns • Multiply, if necessary, to create opposite coefficients for one variable. • Add the equations. • Substitute the value to solve for the other variable. • Check

  26. EXAMPLE 1 (continued) (-1, 3)

  27. EXAMPLE 2 (4, 0) 4x + 3y = 16 2x – 3y = 8

  28. Solve Systems of Equations by Substitution

  29. Steps • One equation will have either x or y by itself, or can be solved for x or y easily. • Substitute the expression from Step 1 into the other equation and solve for the other variable. • Substitute the value from Step 2 into the equation from Step 1 and solve. • Your solution is the ordered pair formed by x & y. • Check the solution in each of the original equations.

  30. Solve by Substitution 1. x = -4 3x + 2y = 20

  31. Solve by Substitution 2. y = x - 1 x + y = 3

  32. Graphing Inequalities

  33. Steps • Get in slope-intercept form • Determine solid or dashed line • Determine whether to shade above or shade below the line (Test Points) • If the test point is true, shade the half plane containing it. • If the test point is false, shade the half plane that does NOT contain the point

  34. Symbols Shade above > ≥

  35. Symbols Shade Below < ≤

  36. Symbols Solid Line ≤ ≥

  37. Symbols Dashed Line < >

  38. Example 1

  39. Example 2

  40. Example 3

  41. Example 3 Solution

  42. Example 4

  43. Example 4 Solution

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