The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of...

The Game of Algebraor

The Other Side of Arithmetic

The Game of Algebraor

The Other Side of Arithmetic

© 2007 Herbert I. Gross

by Herbert I. Gross & Richard A. Medeirosnext

Lesson 12

Linear

Relationships

Linear

Relationships


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In this lesson, as well as in the next several lessons, we will be discussing

problems that involve constant rates of change.

Most rates of change are not constant. For example, if you drive 100 miles in 2

hours, it is highly unlikely that you drove at a constant rate of 50 miles per hour

for the entire 2 hours.


Prelude

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However, it should be noted that in most real life situations even non-constant

rates of change may be considered to be constant if measured over sufficiently small

intervals of time. For example, a person's life may change dramatically over a period of

twenty years. Yet on a day to day basis, the changes usually are not even noticeable.


Prelude

In any case, since problems involving constant rates are relatively easy to analyze,

they make a good starting point for our algebra course.

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If the rate of change of one quantity with respect to another is constant, we say that the relationship between the two quantities

is linear.


Definition D

For example, the relationship between feet and inches is a linear relationship because the rate of change of inches with respect to

feet is constant (that is, there are always 12 inches per foot).

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On the other hand, the relationship between the length of a square’s side and

the area of the square is not linear. That is, the rate of change of the square’s area with respect to the length of one of

the square’s sides is not constant.


This is easy to see if we make a table that shows how each change in the length (L) of

its side changes the square’s area (A).

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next© 2007 Herbert I. Gross

Length (L) Area (L× L) Rate of Change

1 inch 1 square inch

2 inches 4 square inches3 square inches

per inch


per inch


per inch

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If a relationship is linear, we can always write it in a special form. This form will be

developed during our discussion of the following example…


Suppose you decide to buy pens for your office, and that a box of pens contains 12 pens (the constant rate here is 12 pens per box). To get a cheaper price, you decide to buy the pens by the box.

Since each box contains the same number of pens (12), it is easy to compute how many pens you bought once you know

how many boxes you bought. next

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Namely…

2007 Herbert I. Gross

Using the language of algebra, the above three steps can be conveniently

abbreviated as…

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1. Start with the number of boxes you bought.

2. Multiply by 12.

3. The answer is the number of pens you bought.

p = 12n

n

12n

p

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Of course there may be a difference between the number of pens you bought

and the number of pens you have.


For example, suppose you have 5 pens in the office before you buy the other pens.

Then the total number of pens you will have is 5 more than the number of pens

you bought.

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Therefore, if you now wanted to know the total number of pens (T) you will have, the

previous “recipe” would have to be replaced by…


Translating this “recipe” into an algebraic formula, we have…

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1. Start with the number of boxes you bought.

2. Multiply by 12

3. Add 5.

T = 12n + 5

n

12n

12n + 5

4. The answer is the total number of pens you now have. T

to find the number of pens you bought.

The number of pens you already had.

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Since we don't always have to be talking about the number of boxes (n) and the number of pens (T), we can rewrite the

formula T = 12n + 5 in a more general way.


► To begin with, in order to keep the formula as general as possible, it is traditional to denote the “input” (the

number we start with) by the letter “x” and the “output ”(the answer we get) by the

letter “y”.

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► Secondly, there could have been a different number of pens per box. Hence, rather than 12 (pens/box), we could let m

denote the number we multiply x by.


► Similarly, we could let b denote the number of pens we already had in the

office before buying more.

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So if we now denote the input by the letter x and the output by the letter y,

the generalized formula becomes…


Translating these four steps into the language of algebra, we have…

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1. Start with the input.

2. Multiply by m

3. Add b.

y = mx + b

x

mx

mx + b

4. The answer is the output y

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► Any relationship that can be expressed in the form of y = mx + b is

called linear.(See the appendix at the end of this lesson for a geometric interpretation of why the

term “linear” was chosen.)


► More specifically, if m and b are constants, the expression “mx + b” is

said to be linear in x.

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The important thing to remember about the relationship…

y = mx + b

is, that the rate of change of y with respect to x is determined solely by the value of m and has nothing to do with the value of b.

This may be easier to understand in terms of the number of pens in a box and the

number of pens we already had.© 2007 Herbert I. Gross

Note

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For example, suppose there are still12 pens in each box we buy, but the

number of pens we start with is 8 instead of 5. Then, the formula for the

total number of pens is given by…


Note

T = 12n + 8

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If we compare formulas T = 12n + 5 and T = 12n + 8, we see that the multiplier of n is the same (that is, in each case there are 12 pens in a box), but the value of b has changed (that is, we

have changed the number of pens we had before buying more).


Comparing Formulas

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However, suppose now we change the multiplier of n but leave b alone.

For example, suppose we write that…


T = 24n + 5

This formula tells us that we still started with 5 pens, but that the rate of change of T

with respect to n is now 24. That is, the above formula represents the total number

of pens (T) when there were 24 pens in each box we bought, and we started with 5 pens

which were already in the office.next


The correct definition of linear is that the rate of change of y with respect to x is

constant. However, it is often helpful to write the relationship in the form

y = mx + b; in which case m represents this rate of change.

Reminder

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When b ≠ 0, the average cost per box is not the same as the actual cost per box. For example,

suppose that, in addition to the $3 per box, there is also a $5 “shipping and handling charge”. As

shown in the chart below the average cost per box changes as you buy more boxes…


Special Note on Average Cost

Number of boxes(x)1 $8 $82 $11 $5.50

10 $35 $3.50100 $305 $3.05

Total Cost(3x + 5)

Average Cost/Box[(3x + 5) ÷ x]

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In particular, even though 2 boxes of pens is double the number of pens

in 1 box, the cost of buying 2 boxes ($11) was not twice the cost of buying 1 box ($ 8). More specifically,

as the number of boxes we buy increases, the average cost per box gets closer and closer to $3 but will

always be more than $3.



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. In other words, in the relationship

y = mx + b; y increases by the same amount (m) whenever x increases by 1.

However, if b is not equal to 0, the average change in y with respect to x is

not constant.



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However, if b = 0, then we have a direct proportion between x and y. For example, if the cost per box remains $3, but there is no shipping and handling charge we see that…



Number of boxes(x)1 $3 $32 $6 $3

10 $30 $3100 $300 $3

Total Cost(3x)

Average Cost/Box(3x ÷ x)

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Appendix

A Geometric Interpretation of the Word

“Linear”

© 2007 Herbert I. Grossnext

In many real-life situations we find ourselves saying that…

“A picture is worth a thousand words.”

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Notice, for example…


► how many illustrations (pictures) are included in the directions for assembling

any “do it yourself” project;

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► or how many times you may look at a map (a picture) before you describe in

words what route a person should follow;


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► or how many times we use geometric language to describe arithmetic or

algebraic concepts. For example, when the temperature is increasing, we often

describe it by saying that temperature is rising.

► or when we want to multiply a number by itself (arithmetic), we say

“square (geometric shape) the number”.

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An ordinary ruler is an excellent illustration of how we combine arithmetic

and geometry. In effect, a ruler is a straight line (which is a geometric

concept), along which we mark off certain points (also a geometric concept); but then we give these points numerical

names. So in essence the ruler serves as a number line.


The Number Line

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The concept of a linear relationship is easier to visualize in terms of a graph.


Note

In essence: To construct a graph, we choose two

number lines which are perpendicular to one another.

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► One number line (usually drawn horizontally) is referred to as the

x-axis. We choose the positive direction to be from left to right. In terms of our program or recipe, the x-axis usually

denotes the input.

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► The other number line is then drawn perpendicular to the x-axis, and it is

referred to as the y-axis. We choose the upward direction as being the positive

direction. The y-axis denotes the output.

x-axis

y-axis


► The resulting diagram is often referred to as the xy-plane.

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x

y

The arrows point in the positive direction.

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A road map is an example of this kind of geometric thinking. It indexes streets in

the form of a grid with perhaps the columns labeled alphabetically (that is, A, B, C, etc.),

and the rows labeled numerically (that is, 1, 2, 3, etc.,).


Road Map

So if a location is labeled, for example, B3, you look for it in the sector in which column

B meets row 3.next


Since there are “lots” of points in the xy-plane, how can we direct the person to

locate precisely the point that we are thinking of (such as point P as indicated

below)?

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x

y

P


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One way might be to give a set of directions such as…

“Starting on the x-axis at 0;

x

y

P

0

first go 2 units horizontally to the left on the x-axis, and then go vertically upward 3 units .”

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-2 is referred to as the x-coordinate of P; and 3 is referred to as the y-coordinate of P. The custom is to abbreviate the above by enclosing the x-coordinate followed by the y-coordinate in parentheses. Thus, in the present illustration, point P would be representedby the ordered pair(-2,3). x

y

P

(0,0)(-2,0)

(0,3)(-2,3)

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Notice the importance of order; the point denoted by (-2,3) for which x = -2 and y = 3, is not the same point as the point (3,-2) for which x = 3 and y = -2


x

y

0

(-2,3)

More specifically, (3,-2) denotes the point you arrive at when after starting at 0, you move 3 units to the right and then 2 units down.

(3,-2)

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The idea of an ordered pair is not necessary when we look at a road map. That is, whether we write B3 or 3B, it still means the sector in

which column B meets row 3.

It's when both coordinates are numbers that order becomes important.

Road Map

http://www.emaps.ca/clipart_catalog.asp?ProdType=3

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This method of representing points by ordered pairs of numbers was invented by Rene Descartes, a French philosopher, theologian, and mathematician who lived in the 16th century. In his honor, the xy-plane is called the Cartesian Plane. It was Descartes’ ambition to find a way of

uniting algebra and geometry, and the Cartesian Plane allowed him to do that.


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In fact, it is the Cartesian Plane that gives us a geometric way to view algebraic relationships. Consider, for example, the linear relationship…


We may view x as the input and y as the output, and then make a chart similar to our

earlier ones.

y = 2x + 3

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Not Just a Road Map

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The chart follows…


x y = 2x + 32x 2x + 31 52 52 74 73 96 94 118 11-5 -7-10 -72/3 41/3

4/3 41/3

(The last two rows are there as a reminder that the input can be any number; not just

a whole number.)next


Thus, we may use the notation (2,7) as an abbreviation for the statement…

“When the input is 2 the output is 7.” In this way the previous chart can be

abbreviated by our talking about the set of ordered pairs…

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(1,5), (2,7), (3,9), (4,11), ( 5,7), (2/3,41/3), etc.

The collection of ordered pairs we get in this way is called the

graph of the relationship.

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If we now identify the ordered pairs

(1,5), (2,7), (3,9), (4,11), (-5,-7), (2/3,41/3), etc.

with the points…

(1,5), (2,7), (3,9), (4,11), (-5, -7), (2/3,41/3 ), etc.

in the Cartesian Plane, we obtain a “picture” of the graph (which is shown

on the next slide). © 2007 Herbert I. Gross

The picture is shown above… next© 2007 Herbert I. Gross

(1,5)

(2,7)

(3,9)

(4,11)

(0,0)

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(-5,-7)


(1,5)

(2,7)

(3,9)

(4,11)

(0,0)

x

y

Looking at the points that we've graphed, it is easy to see that not only are they on the same straight line, but also that the line rises by 2 units for every 1 unit we move to the right.

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When we refer to the graph of the linear equation y = mx + b, we introduce some

new vocabulary. More specifically…


Some New Terminology

♦ Instead of referring to “the output” we talk about the “rise” of the line.

♦ Instead of talking about “the input” we talk about the “run”.

♦ Instead of referring to m (in the equation y = mx + b) as “the rate of change in the output with respect to the input” we talk

about the “slope of the line”. nextnextnext

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In summary, when we are talking about the line whose equation is y = mx + b, we refer to m as the slope of the line, and we define

the slope to be the rate of change of the rise with respect to the run.


The slope is one way to describe the direction of the line. For example, in the

present illustration, it says that if westart at any point on the line, every time we

move 1 unit in the positive direction the line rises by 2 units.

next© 2007 Herbert I. Gross next

Pictorially…

run1

rise2

x

(1,5)

(2,7)

(3,9)

(4,11)

(0,0)

y


While representing equations geometrically is often very helpful, we

should keep in mind that geometric models are limited to 3 dimensions. This

creates problems when a formula contains several variables.

The Limitations of a Geometric Model

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Recall that we identified the ordered pair (x,y) with the point (x,y) in the Cartesian Plane. This identification worked well

because the equation y = mx + b contains only two variables (x and y), and hence,

its graph could be represented in 2-dimensional space (in this case,

the xy-plane).



For this reason we also refer to the equation y = mx + b as being

2-dimensional. Mathematically speaking, the dimension of an equation is the

number of variables it contains. Hence, it makes sense to talk about equations that

have four or more dimensions, even though there is no geometric counterpart

for equations that have more than 3 dimensions.


In our course, we are concentrating on 2-dimensional equations, such as

y = mx + b. However in most real-life formulas, there are more than two

dimensions (that is, more than 2 variables). For example, the formula for determining the area (z) in square inches of a rectangle

whose base is x inches and whose height is y inches is…

z = xynextx

y


In this case, the output is z but there are now two inputs that determine the value

of z, namely x and y. In this case, the graph of the equation would be the set of ordered triplets (x,y,z); and even though it

now requires 3 dimensions, we can still represent the graph geometrically; but the

concept of slope becomes a bit more difficult to describe .


A more significant problem arises if we have to deal with four or more variables in

a formula. For example, a formula that expresses the volume (v) of a rectangular

box in cubic inches is …

v = xyz

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…where x, y and z represent the dimensions of the box in inches.

xy

z


In this case, the algebraic definition of the graph of the formula is 4-dimensional

because the input consists of the 3variables x ,y and z; and the output (v) is the 4th variable. That is, algebraically speaking, the graph of this formula, while still defined to be the set of pairs (input, output), is now

the set of ordered “4-tuples” (x,y,z,v); and in this case the geometric version

of the graph would require a (nonexistent) 4-dimensional, geometric space.


For this reason, our own preference is to emphasize the terminology “rate of change”

rather than to rely too heavily on such visual terms as “rise”, “run”

and “slope”.

Key Point


Summary –

Algebraic/GeometricTerminology and Usage

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If we “personify” the relationship y = 2x + 3 in terms of, say, buying candy bars for $2 each plus an overall $3 shipping and

handling charge, we are saying that for every candy bar we buy, our cost

increases by $2.© 2007 Herbert I. Gross

The fact that all the points in the graph of y = 2x + 3 lie on the same straight line is the reason that we call the relationship

y = 2x + 3 linear.

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The x-axis and y-axis are straight lines, and hence, we view them as being geometric

entities. However, in the spirit of Descartes, we would like to be able to represent all lines (and other curves as well) in the

xy-plane in terms of algebraic equations.


The Cartesian Plane from

an Algebraic Point of View

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With this in mind, let's see what it means for the point (x,y) to be on the x-axis.


If its y-coordinate is positive (that is y > 0), the point (x,y) is above the x-axis, and if its y-coordinate is negative (y < 0), the point is below the x-axis.

x-axis

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y > 0

y < 0

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Thus, if a point (x,y) is on the x-axis, its y-coordinate is 0.


So, for example, the points (2,0), (-3,0), and (3/4,0) are all on the x-axis.

x-axis

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y-axis

(2,0)(-3,0) (3/4,0)

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By analogous reasoning, if x is positive, the point (x,y) will be to the right of the y-axis;


and if x is negative, the point (x,y) will be to the left of the y-axis.

x > 0

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y-axis

x < 0

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Thus, if a point (x,y) is on the y-axis, its x-coordinate is 0.


So, for example, the points (0,2), (0, -3),and (0,3/4) are on all the y-axis.

x-axis

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y-axis

(0,2)

(0,-3)

(0,3/4)

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In a similar way, the line y = 2 means the set of all points (x,y)

in the Cartesian Plane for which y = 2. Another way of saying this is:

“y = 2 is the line that passes through the point (0,2) and is parallel to the

x-axis”.© 2007 Herbert I. Gross

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x

y

(0,0)

(0,2) (3,2)(-3,2)

y = 2

The points (3,2) and (-3,2) are also shown

on this line. next

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More generally:

If c represents any number (positive, negative, or 0), the line

y = c is the line that passes through the point (0,c) and is

parallel to the x-axis.


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Thus, in much the same way that y = 0 is the equation of the x-axis;

x = 0 is the equation of the y-axis.


x-axis

y-axis

x = 0

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x

y

x = 3

(3,0)

(3,2)

(3,-4)

(0,0)

In a similar way, x = 3 is the set of points (x,y) for which x = 3. In other words, in terms of the Cartesian Plane,

x = 3 represents the line that passes through the point (3,0) and is parallel to the y-axis.

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The points (3,2) and (3,-4). are also shown on this line. next

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Again, more generally:

If c represents any number, the line x = c is the line that

passes through the point (c,0) and is parallel to the y-axis.


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This gives us anotherway to describe apoint such as (3,2).

Namely (3,2) is the point that is on bothlines x = 3 and y = 2. In other words (3,2) represents the point at which the vertical line x = 3 and the horizontal line y = 2 intersect.


x

y

0

(3,2)

x = 3

y = 2

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Prior to this discussion x = 3 referred to a number, and now it refers to a line. How are we to tell which way we're viewing it? The

answer is that if our discussion is about the x-axis, then x = 3 is the point P on the x-axis that is 3 units to the right of 0.


Caution

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0 1 2 3

Px-axis

–1


Caution

On the other hand, if our discussion is about the Cartesian Plane (that is, the xy-plane) then x = 3 represents all the points (x,y) for which x = 3.

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x-axis

(0,0) (1,0) (2,0) (3,0)

x = 3

(-1,0)

y-axis

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Thanks to the work of Descartes, we may view geometric figures in terms of algebraic equations and algebraic

equations in terms of geometric figures.


Algebraic/Geometric

For example, the equation y = 2x + 3 can be viewed in the Cartesian Plane as the straight line L that consists of the set of points (x,y) for which y = 2x + 3. This allows us to see quite visually how “linear” and “line” are

related.next

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Conversely, the straight line L that consists of the set of points (x,y) in the

xy-plane for which y = 2x + 3 can be viewed as the algebraic equation

y = 2x + 3. So, for example, to find the point on L whose x-coordinate is 400, we do not have to draw the line to scale and

then locate the point geometrically. Rather we need only replace x by 400 in the equation y = 2x + 3 to determine that

y = 2(400) + 3 = 803. Therefore, the desired point is (400,803).


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The idea of being able to use pictures and equations

interchangeably is often quite helpful.


Key Point

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