Algebraic Thinking

8.2 Equals Relation and Equations

The Equals Relation as a Mathematical Relation

Equality is a relation with foundational logical properties.

Theorem: Properties of Equality Relation

For real numbers a, b, c:

  • Reflexive: $a=a$
  • Symmetric: if $a=b$ then $b=a$
  • Transitive: if $a=b$ and $b=c$ then $a=c$

Definition: Relation

A relation R from set A to set B is a subset of $A\times B$. If $(a,b)\in R$, write $aRb$.

Example: Analyze a relation

Let

$$R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}.$$

Is R reflexive, symmetric, transitive?

Show Solution
  • Reflexive on {1,2,3}: yes, since (1,1), (2,2), (3,3) are all present.
  • Symmetric: yes for off-diagonal pairs included: (1,2) with (2,1), (2,3) with (3,2).
  • Transitive: no. Because (1,2) and (2,3) are in R, transitivity would require (1,3), which is missing.

So R is reflexive and symmetric, but not transitive.

MyOpenMath: MOM: Relation properties

Properties of Equality for Solving Equations

Theorem: Equality Properties

For real a, b, c:

  • Addition property: if $a=b$, then $a+c=b+c$
  • Multiplication property: if $a=b$, then $ac=bc$
  • Addition cancellation: if $a+b=a+c$, then $b=c$
  • Multiplication cancellation: if $ab=ac$ and $a\ne0$, then $b=c$

Theorem: Zero Product Property

If $ab=0$, then $a=0$ or $b=0$.

Example: Solve linear equations

Solve:

  1. $3x+5=11$
  2. $2x-7=3x+5$
Show Solution
  1. Subtract 5: $3x=6$. Divide by 3: $x=2$.
  2. Subtract $2x$ from both sides: $-7=x+5$. Subtract 5: $x=-12$.

Example: Solve quadratic equations

Solve:

  1. $4x^2-16=0$
  2. $(x+1)(x-2)=0$
Show Solution
  1. $4x^2=16\Rightarrow x^2=4\Rightarrow x=\pm2$.
  2. Zero product gives $x+1=0$ or $x-2=0$, so $x=-1$ or $x=2$.
MyOpenMath: MOM: Solving equations with equality properties

Application Problems with Equations

Example: Library-fine equation

Bruno has 5 overdue books at 0.10 dollars per book per day. One book was checked out 7 days earlier than the other 4. Total fine is 8.70 dollars. Find overdue times.

Show Solution

Let x be overdue days for each of the 4 novels. Then astronomy book is x+7 days overdue.

Total fine:

$$0.10[4x+(x+7)]=8.70.$$

So

$$0.10(5x+7)=8.70\Rightarrow 5x+7=87\Rightarrow 5x=80\Rightarrow x=16.$$

Novels: 16 days each. Astronomy book: 23 days.

Example: Newspaper-delivery system

Abby delivers 3 times Bob. Connie delivers 13 more than Abby. Total is 496 papers. Find each amount.

Show Solution

Let Bob = b. Then Abby = 3b. Connie = 3b+13.

Equation:

$$b+3b+(3b+13)=496$$ $$7b+13=496\Rightarrow 7b=483\Rightarrow b=69.$$

So:

  • Bob: 69
  • Abby: 207
  • Connie: 220

Check: 69+207+220=496.

MyOpenMath: MOM: Equation applications