# Algebraic proofs

### Algebraic proofs

#### Lessons

Let $a,b,$ and $c$ be real numbers. Then here are some of the properties of equality:
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
• 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 $\frac{y}{2}=3$ , then $y=6$ .

b)
$a=a$

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

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

e)
If $ac=bc$ , then $a=b$ .

f)
If $x=8$ and $8=y$ , then $x=y$ .

• 2.
If $3a=3b$ , then which property is used to justify that $a=b$ ?

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

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

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

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