Algebraic proofs

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  1. Algebraic Proofs Overview:
  2. Properties of Equality for Real Numbers
  3. Two-Column Proof Example
  1. Understanding the Properties of Equality
    State which property was used in each statement:
    1. If y2=3\frac{y}{2}=3 , then y=6y=6 .
    2. a=aa=a
    3. If 2x+3=52x+3=5, then 2x=22x=2.
    4. If 3(5x+1)=23(5x+1)=2, then 15x+3=215x+3=2
    5. If ac=bcac=bc , then a=ba=b .
    6. If x=8x=8 and 8=y8=y , then x=yx=y .
  2. If 3a=3b3a=3b , then which property is used to justify that a=ba=b ?
    1. If 3(a+b)=73(a+b)=7 , then which property is used to justify that 3=7a+b3=\frac{7}{a+b} ?
      1. Two Column Proofs
        Prove that if 2(4x+1)=102(4x+1)=10 , then x=1x=1 . Use the two column-proof method
        1. Prove that if 15=2(x+5)+3x515=2(x+5)+3x-5 , then x=2x=2 . Use the two column-proof method.
          1. Prove that if y3+3y4=6 \frac{y}{3} +3y-4=6 , then y=3y=3 . Use the two column-proof method.
            Topic Notes
            Let a,b,a,b, and cc be real numbers. Then here are some of the properties of equality:
            Reflexive Property: For every number aa, then a=aa=a.
            Symmetric Property: For all numbers aa and bb , if a=ba=b , then b=ab=a .
            Transitive Property: If a=ba=b and b=cb=c , then a=ca=c .
            Substitution Property: If a=ba=b , then bb can be substituted for aa in any equation.
            Addition Property: If a=ba=b , then a+c=b+ca+c=b+c .
            Subtraction Property: If a=ba=b , then a?c=b?ca?c=b?c .
            Multiplication Property: If a=ba=b , then ac=bca\cdot c=b\cdot c .
            Division Property: If a=ba=b , then ac=bc\frac{a}{c}=\frac{b}{c} .
            Distribution Property: a(b+c)=ab+aca(b+c)=ab+ac

            When you solve an equation, you will want to use to the two-column proof. For example, if you want to show that x=1x=1 for the equation 2(x+1)+1=52(x+1)+1=5 , then it will look like this:
            Statements Reasons
            2(x+1)+1=52(x+1)+1=5 Given
            2x+2+1=52x + 2 + 1 =5 Distributive Property
            2x=22x=2 Subtraction Property
            x=1x=1 Division Property