Page 183 - ArithBook5thEd ~ BCC

7
Solution. At some point it will be necessary to multiply both sides by the reciprocal of − .But it is
8
a mistake to do that immediately. If we first subtract 5 from both sides, to isolate the linear term, the
computations are shorter.
7
26 = 5 − x
8
− 5 − 5
7
21 = − x
8
8
" #
Now we multiply both sides by −
7
8 " 8 #" 7 #
− · 21 = − − x
7 7 8
8 ✚❃ 3 " ✁ 8 #" ✁ 7 #
− · ✚ 21 = − − x
✚ ✚ 7 ✁ 7 ✁ 8
−24 = x

The check is left to you.

Alinear equation given in the standard form

ax + b = c (a ̸=0)

is solved in two steps:

• first, subtract b from both sides;

• second, divide both sides by a.

Note that this is a reversal of the order of operations: in solving for x we “undo” the operations on
x, from last to first.

Example 231. Solve the equation −x +5 = 12.

Solution. Note that −x means −1x. Following the two step procedure, we first subtract 5 from both
sides, and then divide both sides by −1.

−x +5 = 12
−x = 7 (subtracting 5 from both sides)
x = −7 (dividing or multiplying both sides by −1)

Note that dividing by −1is the same as multiplying by −1, since −1 is its own reciprocal!

1 1
Example 232. Solve 3x + = .
2 2

Page 183