Algebraic proofs

Algebraic proofs

Lessons

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
  • Introduction
    Algebraic Proofs Overview:
    a)
    Properties of Equality for Real Numbers

    b)
    Two-Column Proof Example


  • 1.
    Understanding the Properties of Equality
    State which property was used in each statement:
    a)
    If y2=3\frac{y}{2}=3 , then y=6y=6 .

    b)
    a=aa=a

    c)
    If 2x+3=52x+3=5, then 2x=22x=2.

    d)
    If 3(5x+1)=23(5x+1)=2, then 15x+3=215x+3=2

    e)
    If ac=bcac=bc , then a=ba=b .

    f)
    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 ?

  • 3.
    If 3(a+b)=73(a+b)=7 , then which property is used to justify that 3=7a+b3=\frac{7}{a+b} ?

  • 4.
    Two Column Proofs
    Prove that if 2(4x+1)=102(4x+1)=10 , then x=1x=1 . Use the two column-proof method

  • 5.
    Prove that if 15=2(x+5)+3x515=2(x+5)+3x-5 , then x=2x=2 . Use the two column-proof method.

  • 6.
    Prove that if y3+3y4=6 \frac{y}{3} +3y-4=6 , then y=3y=3 . Use the two column-proof method.