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DESCRIBING MATHEMATICAL SYSTEM PROVE THAT IF A=B, AND C=D, THEN A+C = B+D. COMPLETE THE TABLE BELOW PROVE THAT IF a=b, and c=d, then a+c = b+d. Complete the table below STATEMENT REASON 1. a=b Given 2. a + c = b + c Add both side by c 3. c =d Given 4. c + b = d + b Add both side by b 5. a + c = b + d Addition Property of Equality GIVEN: X, Y AND Z ARE REAL NUMBERS AND X>Y PROVE: X + Z > Y + Z Given: x, y and z are real numbers and x>y Prove: x + z > y + z STATEMENT 1. x > y REASON Given 2. x + z > y + z Addition axiom GIVEN: -4 = M PROVE: M = -4 Given: -4 = m Prove: m = -4 STATEMENT 1.-4 = m 2. -4 + 4 = m + 4 3. 0 = m + 4 4. 0 – m = m + 4 -m 5. –m = 0 + 4 6. –m = 4 7. (-1)(-m) = (-1)(4) 8. m = -4 REASON Given Addition Axiom Existence of additive inverse Addition axiom Existence of additive inverse Additive identity Multiply both side by -1 Multiplication axiom GIVEN: 4(2X-3) + 16 = 5X + 37 PROVE: X = 11 Given: 4(2x-3) + 16 = 5x + 37 Prove: x = 11 STATEMENT REASON 1. 4(2x-3) + 16 = 5x + 37 Given 2. 8x – 12 + 16 = 5x + 37 Distributive Property 3. 8x + 4 = 5x + 37 Combined like terms 4. 8x + 4 – 4 = 5x + 37 - 4 Addition axiom 5. 8x = 5x + 33 Existence of additive inverse 6. 8x -5x = 5x + 33 – 5x Addition axiom 7. 3x = 33 Existence of additive inverse 8. 1/3 (3x) = (33)1/3 Existence of multiplicative inverse 9. x = 11 Simplify GIVEN: 7B – 25 = 2B PROVE: B = 5 Given: 7b – 25 = 2b Prove: b = 5 STATEMENT REASON 1. 7b – 25 = 2b Given 2. 7b – 25 + 25 = 2b + 25 Addition axiom 3. 7b = 2b + 25 Existence of additive inverse 4. 7b – 2b = 2b + 25 -2b Addition axiom 5. 7b – 2b = 25 Existence of additive inverse 6. 5b = 25 Combine like terms 7. 1/5(5b) = 1/5 (25) Existence of multiplicative inverse Simplify 8. b = 5 FIGURE IT OUT 2 (-a + 5) = (2)(-a) + (2)(5) 2/5 . 1 = 2/5 (X + 5)+2 = X+(5+2) ¼(4X) = X ( x + y) + z = z + (x+ y) FIGURE IT OUT 2 (-a + 5) = (2)(-a) + (2)(5) DISTRIBUTIVE PROPERTY 2/5 . 1 = 2/5 EXISTENCE OF MULTIPLICATIVE IDENTITY AXIOM (X + 5)+2 = X+(5+2) ASSOCIATIVE AXIOM ¼(4X) = X EXISTENCE OF MULTIPLICATIVE INVERSE ( x + y) + z = z + (x+ y) COMMUTATIVE AXIOM 1. _______________ ARE TERMS THAT DO NOT HAVE CONCRETE DEFINITION BUT CAN BE DESCRIBED. ON THE OTHER HAND, 2. ______________ REQUIRE DEFINITION. THERE ARE STATEMENTS ASSUMED TO BE TRUE EVEN WITHOUT PROOF WHICH WE CALLED AS AXIOMS OR POSTULATES. HOWEVER, THE TWO HAS DISTINCTION IN SUCH A WAY THAT 3.____________ ARE OFTEN USED IN GEOMETRY WHILE THE 4. ___________ ARE USED IN ALL AREAS OF MATHEMATICS. WHEN THE STATEMENT SHOWS EVIDENCES OR PROVEN TO BE TRUE, WE CALL IT AS 5. _________________. UNDEFINED TERMS DEFINED TERMS AXIOMS POSTULATES THEOREMS