Type in the missing values to complete the simplified form of the expression 4x –4 + 3y + 6 –2x + 3y.

FIRST-DEGREE EQUATIONS AND INEQUALITIES IN TWO VARIABLES

The language of mathematics is particularly effective in representing relationships between two or more variables. As an example, let us consider the distance traveled in a certain length of time by a car moving at a constant speed of 40 miles per hour. We can represent this relationship by

1. A word sentence:

The distance traveled in miles is equal to forty times the number of hours traveled.

2. An equation:

d = 40r.

3. A tabulation of values.

4. A graph showing the relationship between time and distance.

We have already used word sentences and equations to describe such relationships; in this chapter, we will deal with tabular and graphical representations.

7.1 SOLVING EQUATIONS IN TWO VARIABLES

ORDERED PAIRS

The equation d = 40f pairs a distance d for each time t. For example,

if t = 1, then d = 40

if t = 2, then d = 80

if t = 3, then d = 120

and so on.

The pair of numbers 1 and 40, considered together, is called a solution of the equation d = 40r because when we substitute 1 for t and 40 for d in the equation, we get a true statement. If we agree to refer to the paired numbers in a specified order in which the first number refers to time and the second number refers to distance, we can abbreviate the above solutions as (1, 40), (2, 80), (3, 120), and so on. We call such pairs of numbers ordered pairs, and we refer to the first and second numbers in the pairs as components. With this agreement, solutions of the equation d - 40t are ordered pairs (t, d) whose components satisfy the equation. Some ordered pairs for t equal to 0, 1, 2, 3, 4, and 5 are

(0,0), (1,40), (2,80), (3,120), (4,160), and (5,200)

Such pairings are sometimes shown in one of the following tabular forms.

Step-by-step explanation: