Find the missing component so that the ordered pair is a solution to
2x + y = 4
a. (0,?)
b. (1,?)
c. (2,?)
Solution
if x = 0, then 2(0) + y = 4
y = 4
if x = 1, then 2(1) + y = 4
y = 2
if x = 2, then 2(2) + y = 4
y = 0
The three pairings can now be displayed as the three ordered pairs
(0,4), (1,2), and (2,0)
or in the tabular forms
EXPRESSING A VARIABLE EXPLICITLY
We can add -2x to both members of 2x + y = 4 to get
-2x + 2x + y = -2x + 4
y = -2x + 4
In Equation (2), where y is by itself, we say that y is expressed explicitly in terms of x. It is often easier to obtain solutions if equations are first expressed in such form because the dependent variable is expressed explicitly in terms of the independent variable.
For example, in Equation (2) above,
if x = 0, then y = -2(0) + 4 = 4
if x = 1, then y = -2(1) + 4 = 2
if x = 2 then y = -2(2) + 4 = 0
We get the same pairings that we obtained using Equation (1)
(0,4), (1,2), and (2,0)
We obtained Equation (2) by adding the same quantity, -2x, to each member of Equation (1), in that way getting y by itself. In general, we can write equivalent equations in two variables by using the properties we introduced in Chapter 3, where we solved first-degree equations in one variable.
Equations are equivalent if:
The same quantity is added to or subtracted from equal quantities.
Equal quantities are multiplied or divided by the same nonzero quantity.
Example 2
Solve 2y - 3x = 4 explicitly for y in terms of x and obtain solutions for x = 0, x = 1, and x = 2.
Solution
First, adding 3x to each member we get
2y - 3x + 3x = 4 + 3x
2y = 4 + 3x (continued)
Now, dividing each member by 2, we obtain
In this form, we obtain values of y for given values of x as follows:
In this case, three solutions are (0, 2), (1, 7/2), and (2, 5).
FUNCTION NOTATION
Sometimes, we use a special notation to name the second component of an ordered pair that is paired with a specified first component. The symbol f(x), which is often used to name an algebraic expression in the variable x, can also be used to denote the value of the expression for specific values of x. For example, if
f(x) = -2x + 4
where f{x) is playing the same role as y in Equation (2) on page 285, then f(1) represents the value of the expression -2x + 4 when x is replaced by 1
f(l) = -2(1) + 4 = 2
Similarly,
f(0) = -2(0) + 4 = 4
and
f(2) = -2(2) + 4 = 0
The symbol f(x) is commonly referred to as function notation.
Example 3
If f(x) = -3x + 2, find f(-2) and f(2).
Solution
Replace x with -2 to obtain
f(-2) = -3(-2) + 2 = 8
Replace x with 2 to obtain
f(2) = -3(2) + 2 = -4
