Respuesta :
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.
In any particular equation involving two variables, when we assign a value to one of the variables, the value for the other variable is determined and therefore dependent on the first. It is convenient to speak of the variable associated with the first component of an ordered pair as the independent variable and the variable associated with the second component of an ordered pair as the dependent variable. If the variables x and y are used in an equation, it is understood that replace- ments for x are first components and hence x is the independent variable and replacements for y are second components and hence y is the dependent variable. For example, we can obtain pairings for equation
by substituting a particular value of one variable into Equation (1) and solving for the other variable.
Example 1
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.
Hope this helped
Please give me Brainliest
Answer:
[tex]y=-2x+3[/tex]
[tex]y=x+4[/tex]
Step-by-step explanation:
Tell whether each equation can be written in the y= mx + b.
[tex]y=8 - x^2[/tex]
In this given equation , second term is x^2. y=mx+b has only x term. there is no x^2 term
So [tex]y=8 - x^2[/tex] is not written in y=mx+b form
[tex]y= 4 + x[/tex]
Arrange the terms in y=mx+b form
[tex]y=x+4[/tex]
[tex]y= 3 - 2x[/tex]
Arrange the terms in y=mx+b form
[tex]y=-2x+3[/tex]
