# Algebraic proofs

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##### Intros

###### Lessons

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##### Examples

###### Lessons

**Understanding the Properties of Equality**

State which property was used in each statement:- If
$3a=3b$
, then which property is used to justify that
$a=b$
?
- If
$3(a+b)=7$
, then which property is used to justify that
$3=\frac{7}{a+b}$
?
**Two Column Proofs**

Prove that if $2(4x+1)=10$ , then $x=1$ . Use the two column-proof method- Prove that if
$15=2(x+5)+3x-5$
, then
$x=2$
. Use the two column-proof method.
- Prove that if
$\frac{y}{3} +3y-4=6$
, then
$y=3$
. Use the two column-proof method.

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###### Topic Notes

Let
$a,b,$
and
$c$
be real numbers. Then here are some of the properties of equality:

When you solve an equation, you will want to use to the two-column proof. For example, if you want to show that $x=1$ for the equation $2(x+1)+1=5$ , then it will look like this:

**Reflexive Property:**For every number $a$, then $a=a$.**Symmetric Property:**For all numbers $a$ and $b$ , if $a=b$ , then $b=a$ .**Transitive Property:**If $a=b$ and $b=c$ , then $a=c$ .**Substitution Property:**If $a=b$ , then $b$ can be substituted for $a$ in any equation.**Addition Property:**If $a=b$ , then $a+c=b+c$ .**Subtraction Property:**If $a=b$ , then $a?c=b?c$ .**Multiplication Property:**If $a=b$ , then $a\cdot c=b\cdot c$ .**Division Property:**If $a=b$ , then $\frac{a}{c}=\frac{b}{c}$ .**Distribution Property:**$a(b+c)=ab+ac$When you solve an equation, you will want to use to the two-column proof. For example, if you want to show that $x=1$ for the equation $2(x+1)+1=5$ , then it will look like this:

Statements |
Reasons |

$2(x+1)+1=5$ | Given |

$2x + 2 + 1 =5$ | Distributive Property |

$2x=2$ | Subtraction Property |

$x=1$ | Division Property |

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