Page 17 - Schaum's Outline of Theory and Problems of Applied Physics

2 USEFUL MATH [CHAP. 1

We proceed as follows:
1. Subtract y from x to give

x − y = 15 − 3 = 12
2. Divide x − y by z to give
x − y 12
= = 3
z 4
3. Multiply (x − y)/z by5togive

x − y
5 = (5)(3) = 15
z
4. Add w to 5(x − y)/z to give

x − y
v = 5 + w = 15 + 10 = 25
z
SOLVED PROBLEM 1.3

Examples of multiplication and division:
−16
(−3)(−5) = 15 = 4
−4
10
(2)(−4) =−8 =−2
−5
−24
(−12)(6) =−72 =−6
4
SOLVED PROBLEM 1.4

Find the value of w in the equation
xy
w =
x + y
when x = 5 and y =−6.
Here xy = (5)(−6) =−30 and x + y = 5 + (−6) = 5 − 6 =−1. Hence
xy −30
w = = = 30
x + y −1

EQUATIONS
An equation is a statement of equality: Whatever is on the left-hand side of any equation is equal to whatever is
on the right-hand side. The symbols in an algebraic equation usually cannot have arbitrary values if the equality
is to hold. To solve an equation is to find the possible values of these symbols. The solution of the equation
5x − 10 = 20 is x = 6 because this is the only value of x for which this equation is a true statement.
The algebraic procedures that can be used to solve an equation are all based on the principle that any
operation performed on one side of an equation must be performed on the other side as well. Thus an equation
remains valid when the same quantity is added to or subtracted from both sides or is used to multiply or divide
both sides. Other operations, such as raising to a power or taking a root, also do not change an equality if the
same thing is done to both sides.
Two helpful rules follow from the above principle. First, any term on one side of an equation may be shifted
to the other side by changing its sign. Thus if a + b = c, then a = c − b; and if a − d = e, then a = e + d.