Identities and Transposition

Equations come in two kinds.

An identity is an equation whose two members become exactly alike, term for term, once you carry out the operations. 4 \cdot 5 + 3 \cdot 4 = 8 \cdot 5 - 8 is one: both sides are 32. A literal identity like (a + 3)^2 = a^2 + 6a + 9 is true for every value of its letters. Try a = 5: 8^2 = 25 + 30 + 9, so 64 = 64. Try a = 0: 9 = 9. Sometimes we write \equiv ("is identical with") to stress it: 3a \equiv 2a + a.

An equation of condition (usually just called an equation) is true only for certain values. 4x = x + 12 is true only when x = 4; put in 0 or 3 and it's false. Each such equation asks a question: 3x + 2 = 15 asks "What number, times 3, plus 2, gives 15?"

A value that turns an equation into an identity satisfies it and is called a root. Both 2 and 3 are roots of x^2 - 5x + 6 = 0, since 4 - 10 + 6 = 0 and 9 - 15 + 6 = 0. Solving an equation means finding its roots, and checking means confirming that what you found really is a root.

Transposition

Solving 5k - 4 = 3k + 18 by the axioms, we add 4 and subtract 3k from each member: 5k - 4 + 4 - 3k = 3k + 18 + 4 - 3k The -4 + 4 and 3k - 3k cancel, leaving 5k - 3k = 18 + 4. Compare with the original: -4 vanished from the left and showed up as +4 on the right, and 3k left the right side and reappeared as -3k on the left.

Transposition. A term may be dropped from one member of an equation if the same term, with its sign changed, is written in the other member.

It's really just subtracting that term from both members, done faster. Always combine like terms within a member first, then transpose.

Example: 8x - 5 + 4x + 12 = 13x - 10 - 3x + 29 \begin{array}{rll} 12x + 7 &= 10x + 19 & \text{combine like terms} \\ 12x - 10x &= 19 - 7 & \text{transpose} \\ 2x &= 12 & \text{combine} \\ x &= 6 & \text{divide by 2} \end{array} Check: 48 - 5 + 24 + 12 = 78 - 10 - 18 + 29, or 79 = 79. ✓

If you transpose every term, you find you can change the sign of every term in an equation without breaking it. That's the same as multiplying both members by -1.