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Properties of Real Numbers: Simplifying Expressions

Learn to simplify expressions by applying the properties of identity, equality, distributive, commutative, and associative properties. This lesson covers various properties such as Additive Identity, Multiplicative Identity, Multiplicative Property of Zero, Multiplicative Inverse Property, Reflexive Property, Symmetric Property, Transitive Property, and Substitution Property. Practice problems included.

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Properties of Real Numbers: Simplifying Expressions

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  1. Field Properties and Axioms of Equality Mrs. Hacker Algebra 1 Grade: 8th Algebra 1 SC Standards: A.4 Carry out a procedure using the properties of real numbers (including commutative, associative, and distributive) to simplify expressions. O: SWBAT recognize and use the properties of identity, equality, distributive, commutative and associative properties to simplify expressions.

  2. For any number a, a + 0 = 0 + a. “Identity à gets itself back" General: x + 0 = x 0 + z = z For any number x, x * 1 = 1 * x = x “Identity à gets itself back" General: a *1 = a 1 * q = q For an number r, r*0 = 0*r = 0 “Always returns (or is = to) a 0.” General: a * 0 = 0 0 * x = 0 0 = c*0 Additive Identity Property Ex: 1 3 + 0 = 3 Ex: 2 –7 + 0 = –7 Ex: 3 ½ + 0 = ½ Multiplicative Identity Property Ex.1 -1/3 * 1 = -1/3 Ex.2 7 = 1 * 7 Ex.3 -14 * 1 = -14 Multiplicative Property of Zero Ex.1 1 * 0 = 0 Ex.2 -8 * 0 = 0 Ex.3 ¼ * 0 = 0

  3. Your Turn! Ex.1 Using the additive identity property, solve each equation. a. a + 0 = b. 3 + 0 = c. 0 + x = a 3 x Ex.2 Using the multiplicative property of zero, solve each equation. a. x * 0= b. 3(0) 0 0 Ex.3 Name the multiplicative inverse of each number. c. a d. -1/x a. 5 b. -1/3 -x 1/5 1/a -3

  4. Multiplicative Inverse Property For every non-zero number a/b, where b does not equal 0, there is exactly one number, b/a such that a/b * b/a =1. "Flip the fraction and multiply to get 1" General: x/1 * 1/x = 1 1/a * a/1 = 1 r/s * s/r = 1 For any number a, a = a. General: x = x, -r = -r “Any number is equal to itself” For any numbers, a and b, if a = b, then b = a "Mirror Image" General: if a + b = c, then c = a + b Ex. 1 ½ * 2/1 = 1 Ex. 2 -4/1 * -1/4 = 1 Ex. 3 7/9 * 9/7 = 1 Reflexive Property Ex.1 3 = 3 Ex.2 - ½ = - ½ Ex. 3 -75 = -75 Symmetric Property Ex.1 if 3 + 5 =8, then 8 = 3 + 5 Ex.2 if ½ + ¼ = ¾ , then ¾ = ½ + ¼

  5. Your Turn! Ex.4 Name the multiplicative inverse of each number or variable. Assume that no variable equals zero. a. 5 b. x c. 2/3 1/5 1/x 3/2 Ex.5 Using the reflexive property, solve each equation. a. a = b. 3 = c. xy = a 3 xy Ex.6 Using the symmetric property, solve each equation. a. x + y = b. if 7 + 1 = 8, then y + x 8 = 7 + 1

  6. Transitive Property For any numbers x, y, and z, if x = y and y = z, then x = z "If 1st = 2nd, and 2nd = 3rd, then 1st = 3rd” General: If a = b and b = c then a = c If x = q + r and q + r = y, then x = y If a = b, then a may be replaced by b in any expression "Think replacement" General: x = y, x + 7 = y + 7, x/3 = y/3 Ex.1 if 2 + 4 = 6 and 6 = 3 + 3, then 2 + 4 = 3 + 3 Ex.2 if -7 + 4 = -3 and -3 = 1 – 4, then -7 + 4 = 1 – 4 Ex.3 if 10 = 2(5) and 2(5) = 1(10), then 10 = (1)(10) Substitution Property Ex.1 (1+2)(3) = 3(3) Ex.2 9(4+2) = 9(6) Ex.3 14 + 2 = 16

  7. Your Turn! Ex.7 Using the transitive property, solve each equation. a. If 4 * 2 = 8 and 8 = 6 + 2, then 4 * 2 = 6 + 2 b. 12 = (-3)(-4) and (-3)(-4) = (6)(2), then 0 12 = (6)(2) Ex.8 Using the substitution property, solve equation. c. If a = 7, then 5a = a. 22 b. 10 ÷ 5 4 35 -3

  8. term - a number, a variable, or a product or quotient of numbers and variables. coefficient - the number in front of the term. like terms - terms that have the same variables raised to the same exponent (or power), but could have different coefficients. 3x2y term

  9. Examples of terms: Are they like terms or unlike terms? Pull Pull Pull Pull Pull Pull Pull ·3x and -7x like unlike ·6w and 8y ·8y2, 2y2 and -3y2 like like ·5, 89 and 100 unlike ·4x and 4x2 unlike ·5 and 8x ·x3y4 and ½x3y4 like

  10. 3y x 2x2y 4 3x2y 7x x2y 5 7y2 6x 7y 10x 6x2 7x 10x2 Identify the like terms in the following expressions: Ex.93y + 2x2y - x + 4 + 3x2y - 5 + 7x - x2y Ex.10 6x + 6x2 + 7x + 7y + 7y2+ 10x + 10x2

  11. To combine like terms: ·Figure out which terms are like terms. ·Add or subtract the coefficients (# in front.) ·Attach the variable or variables at the end.

  12. Simplify each expression: Ex.11 7x + 9x + 8 + 13 Ex.12 19x2 + 7y2 + 21x2 16x + 21 40x2 + 7y2 Ex.13 9a + 60a -10 Ex.14 -7n - 21 - 8 - 64n 69a - 10 -71n - 29

  13. For any numbers a, b, and c: a(b+c) = ab + ac and (b+c)a = ba + ca a(b-c) = ab – ac and (b-c)a = ba – ca "Distribute one number to the rest" For any numbers a and b, a + b = b + a and ab = ba General: a(b + c) = a(c + b) and xyz = yxz "Change order" For any numbers x, y, and z, (x+y) + z = x +(y+z) and (xy)z = x(yz). "Same order, but the parenthesis go in a different spot" General: (qr)s=q(rs) 7x + (3y - 2z) = (7x + 3y) - 2z Distributive Property Ex.1 x(8+3) = 8x + 3x Ex.2 2(4+t) = 8 + 2t Ex.3 8(3m + 6) = 24m + 48 Commutative Property DOES NOT work for division or subtraction! Ex.1 7 + 6 = 6 + 7 Ex.2 3(5) = 5(3) Ex.3 4(2*7) = 4(7*2) Associative Property Ex.1 (4 + 6) + 8 = 4 + (6 + 8) Ex.2 (2c + 6) + 10 = 2c + (6 + 10) Ex.3 7(11*3) = (7*11)3

  14. Your Turn! Ex.15 Using the distributive property, solve each equation. a. 2(2 + x) b. y(a - 3) c. x(1 + y) 4 + 2x ay -3y x + xy Ex.16 Using all the properties, solve each equation. b. 4a + 2b + a a. 3(4x + y) + 2x 5a + 2b 14x + 3y d. 4y4 + 3y2 + y4 c. 4 + 6(ac + 2b) + 2ab 4 + 8ac + 12b 5y4 + 3y2

  15. Know your properties or else!

  16. Math Hoops

  17. Match your Properties again!

  18. Credits: Games: http://www.quia.com/hm/26073.html http://www.math-play.com/math-basketball-properties-of-multiplication/math-basketball-properties-of-multiplication.html Images: http://math.phillipmartin.info/math_properties.htm http://www.wordle.net/

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