Algebra Know-It-ALL

a− 8 x=b

− 6 a+ 3 x= 12 b

6 a− 3 x/(bc)=− 24 d

These seven equations can all be rearranged with the morphing laws we already know, so that
x appears alone on the left sides of the equality symbols, and nothing but constants appear on
the right sides. Here are the respective solutions:

x= 1/2

x=−1/2

x=a/3

x= 1/3 −a/15

x=a/8−b/8

x= 2 a+ 4 b

x= 2 abc+ 8 bcd

Are you confused?

Tables 12-11 through 12-17 break down the solution processes for the above equations. Some of the
steps are combined, making the derivations less tedious than those earlier in this chapter. Note that in
the last original equation above (and in Table 12-17), it’s necessary that b≠ 0 and c≠ 0. Also note that
an attempt has been made to put the solutions in elegant form by avoiding sums or differences in the
numerators of fractions, putting fractions in lowest terms, and getting the letters for the constants in
alphabetical order.

Combinations of Operations 199

Table 12-11. Streamlined process for solving

the equation 8x− 4 = 0.

Statements Reasons

8 x− 4 = 0 This is the equation we are given

8 x= 4 Add 4 to each side

x= 4/8 Divide each side by 8

x= 1/2 Put the fraction into lowest terms

Table 12-12. Streamlined process for solving

the equation 18x+ 7 =−2.

Statements Reasons

18 x+ 7 =− 2 This is the equation we are given

18 x=− 9 Subtract 7 from each side

x=−9/18 Divide each side by 18

x=−1/2 Put the fraction into lowest terms