Page 182 - ArithBook5thEd ~ BCC

Equations which have the same solution are called equivalent. As the previous examples show, a
strategy for finding the solution of an equation is to systematically transform it into simpler equivalent
equations, until we arrive at an equation like x =8, or −45 = z,whose solution is obvious.

To find the solution (solve) a linear equation in one variable,
we can do one or both of the following:

• add the same number to both sides of the equation;

• multiply both sides of the equation by the same non-
zero number.

We can replace “add” by “subtract,” and “multiply” by “divide,”if convenient. (This is because
subtracting a number is the same as adding the opposite number, and dividing by a nonzero number is
the same as multiplying by the reciprocal number.)
Example 229. Solve the equation 2x +4 = −10 and check that the solution is correct.

Solution. First subtract 4 from both sides:

2x +4 = −10
−4 −4
2x = −14.

Then divide both sides by 2:

2x = −14
✟✯
−14
✚ ✚ 2 · x ✟ ✟ −7
=
✚ ✚ 2 ✚❃ 1
✚ 2
x = −7.
The solution is −7. To check, we substitute −7for x in the original equation, and see if a true statement
results.

2x +4 = −10
?
2(−7) + 4 = −10
?
−14 + 4 = −10
−10 = −10 a true statement

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Example 230. Solve the equation 26 = 5 − x and check the solution.
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