Variable as Generalization Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Show that

a + b = b + a

holds for

a = 5, b = 3

and for

a = -2, b = 7

1
Test $a = 5, b = 3$ : $5 + 3 = 8$ and $3 + 5 = 8$ . Equal ✓
2
Test $a = -2, b = 7$ : $-2 + 7 = 5$ and $7 + (-2) = 5$ . Equal ✓
3
The equation holds for both pairs because it is true for ALL values of $a$ and $b$ .

Answer

Both cases verify

a + b = b + a

Here

a

and

b

are not unknowns to solve for—they represent any numbers whatsoever. The statement

a + b = b + a

(the commutative property) is a generalization that works for all real numbers.

About Variable as Generalization

A variable standing for any arbitrary member of a specified set, used to express statements that hold universally.

Explain why [formula] is true for every integer [formula].

Is [formula] a generalization or an equation to solve?

Write a general formula for the sum of the first [formula] positive integers.