Download Section 3.1

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Chapter 3 - Lines
Section 3.1
Graphing
When we solved equations that looked like 3x-2=13, we got a solution like
x=5, but what does that really mean? We have followed an algorithm to arrive
at the proper location, but the reader is reminded that the whole purpose of
manipulating equations is to find numbers for x that we can stick in and make
a true statement. If we stick in 5 for x in this equation, we get 3(5) - 2 = 13,
which is true. There is no other number which will do this. We call this a
solution to the equation.
In that kind of equations we found a number for x that made the statement
true, and sometimes we could even guess what would work without really
using any formulas or steps. This process becomes a bit helpful when studying
the next type of equation:
3x+2y=5
In this type of equation there is an x and a y to find numbers for. The
solution to this equation will not be a single number as it was in the earlier
cases, but a pair of numbers. The answers will look something like (3,-2),
which means that we will stick in 3 for x and -2 for y. If you stick those in,
the equation becomes:
3(3)+2(-2) = 5
9-4=5
Woo Hoo! It works! We found a solution, and we don’t even know what we
are doing yet. Let’s see if there is another one. Try the following pairs of
numbers in the equation to see if they also work:
(1,1)
(3,2)
(-1,4)
(5,-2)
(5,-5)
3(1)+2(1)=5
3(3)+2(2) =5
3(-1)+2(4)=5
3(5)+2(-2)=5
3(5)+2(-5)=5
solution
nope
solution
nope
solution
If you try all of these, you will realize that some of them work as
solutions and some of them don’t. In any case, you should be able to realize
that there are a whole lot of solutions, tons of them! One way to get them is to
keep guessing. When you get tired of that there is an algorithm that might
make things a little easier. If we pick a number for x and stick it in, then we
will have an equation that we can solve for y. For example, if we say in this
example we want x to be 7, we stick it in to get:
3(7)+2y=5
21+2y=5
2y=-16
y=-8
Which means that when x is 7, y will be -8, or
in other words, the pair (7,-8) is a solution. What would we get if we made
y=9. The equation would be:
3x+2(9)=5
2
Solving for x, we get x= -5, so the pair (-5,9 is a solution).
Now we can get so many solutions this way that it doesn’t pose a problem to
find one anymore. Since there are so many, the question arises, “Are there are
patterns in the solutions to these equations.” Well, of course there are. This is
math! The solutions are pairs, which we can stick on a graph. If we plot the
ones that we have already found to the problem we are using we get this: (3,2), (1,1), (-1,4), (5,-5), (7,-8)
You will notice that all of the solutions
are in a straight line. If we connect them,
we get all of the solutions for the
equation. It is important to realize that if
we draw the line that connects the dots,
all of the points on that line are
solutions. The problems will simply ask
you to graph the line 3x+2y=5 or
something similar.
.
.
.
The correct answer to “Graph the line
3x+2y=5”, is then the graph at the left.
.
For the next one, find four points on the line and then graph it:
y = 14 x – 2
x y
when x = 4 we have y = 14 (4) – 2 which means y = -1.
4
when x = 4 we have y = 14 (0) – 2 which means y = -2.
0
0 when y = 0 we have 0 = 14 (x) – 2 which means x = 8.
3 when y = 3 we have 3 = 1 (x) – 2 which means x = 20.
4
x y
4 -1
0 -2
8 0
20 3
Notice that we really only need two points to get the pattern. For
convenience, we often select 0 for x, and then 0 for y.
The table completely filled out looks like this:
and the graph like this:
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When x is 0 the point is on the y-axis. Likewise, when y is 0 the point is on the
x-axis. In the previous example, the point (0,-2) lies on the y-axis and is called
the y-intercept; the point (8,0) lies on the x-axis and is called the x-intercept.
An x-intercept happens
when y is zero, and a yintercept happens when x
is zero.
Here is another example.
Graph the line 4x + 3y = 8; find the x and y intercepts.
We start by finding the x- and y-intercepts with a table that looks like:
x y
0
0
Then fill it out by plugging in 0 for x and getting 4(0) + 3y = 8
3y = 8
y = 83
When we plug in 0 for y we get: 4x + 3(0) = 8
4x = 8
x=2
so we have the table:
x y
0 83
2 0
and the graph:
There are a couple of particular kinds of lines that may give you a bit of
trouble when you first see them. Your first reaction when asked to graph the
line:
x=4
is probably something like, “Hey, where is the y?” or, “How do I do
that? It looks different.”
Relax, these kind are actually a bit easier than the other ones. Watch:
What is x when y is 7? Answer: 4
The points (4,7) (4,0)
What is x when y is 0? Answer: 4
and (4,-3) are part of
What is x when y is -3? Answer: 4
the line and help us
graph it.
Do you see how nice that is? Since y is not in
the equation it can be anything it wants to be, but x is always 4. The graph is as
follows:
4
Here is the line x = 4; notice that it is vertical and
hits where x is 4.
For future reference you can remember that all
equations that only have an x will be vertical.
The other special case that may seem difficult at first looks like:
y = -2
But I think you can see that it will be very similar to the previous example:
What is y when x is 0? Answer: -2
What is y when x is 5? Answer: -2
What is y when x is -3? Answer -2
The points (0,-2) (5,-2)
and (-3,-2) are part of the
line and help us graph it.
See how slick that is?! The graph is as follows:
Here is the line y = -2; notice that it is
horizontal and hits where y is -2.
All equations with just a y in them will be
horizontal lines.
Section 3.2
Slope
Now that we can graph any lines,
there is one particular property of lines that is most useful. We
introduce this by bringing to mind a familiar road sign. This sign
warns of steepness, but take a look at what it is really saying.
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7% means the fraction 100
.The interpretation of the sign
means that the road falls vertically 7 feet for every 100
feet that you travel horizontally. In this way the highway
department uses fractions to denote the steepness of
roads. We are going to do the same thing with the steepness
7%
of lines.
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When we have a couple of points on the graph we can find the steepness
between them. Here are a couple of examples.
1st Example
nd
2 Example
3rd Example
The steepness of the line between point A(-2,1) and B(3,3) is found
by taking how much it changes up and
down (distance between 1 and 3 = 2)
C (1,7)
over how much it changes left and
right(distance between -2 and 3 = 5).
That makes a steepness of 25 . The
B (3,3)
A (-2,1)
name for steepness is slope, and the
symbol is m (as in a mountain). We
would write that m = 25 .
D (5,-3)
The slope of the line through A(-2,1)
and C(1,7) would be 6 (the distance
from 1 to 7) over 3 (the distance from
-2 to 1. We would write m = 63 , or in
other words m = 2.
The slope of the line through A(-2,1) and D(5,-3) would be -4 (the distance
from 1 down to -3) over 7 (the distance from -2 to 5); m = - 74 .
There are some properties that you should start to see from these
examples.
1. Bigger numbers for slope correspond to steeper lines.
2. Positive slopes head up as you go to the right.
3. (Opposite of #2) Negative slopes will head down as you go to the
right.
As a note: You should
realize that the
subtraction may
happen in the opposite
direction but will still
give the same slope.
Example #1 would
look like this:
1− 3
−2 2
=
=
−2−3 −5 5
In the first example we obtained the 2 as the distance from 1 to 3. What
operation finds distance? Answer: Subtraction. Aha! Seeing that, we can start
to see a pattern in how to find slopes a little more quickly. Let’s look at those
three examples, using subtraction this time:
1st Example:
3 −1
2
=
3 − −2 5
2nd Example:
7 −1 2
=
1 − −2 5
3rd Example:
− 3 −1
4
=−
5 − −2
7
Now, see if you can find the slope between two general points:
Point 1 and point 2 with coordinates that we don’t know.
We would like to call them both just (x,y), but then subtraction would
give us zero. This is a good place to introduce you to how subscripts can be
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very helpful. We will call point #1 (x1,y1) showing that the x and the y come
from the 1st point. Similarly we will call point #2 (x2,y2).
Now you can find the slope just like we did in the previous examples:
y − y1
m= 2
x 2 − x1
Voila! You have just created the formula for finding slope between two points.
Practice using it quite a bit until it almost becomes natural. Memorize it!
Sometimes formulas are written in a few different ways. Here are some
of the others:
m=
y1 − y 2
x1 − x 2
m=
ychange
xchange
m=
rise
run
m=
∆y
∆x
They all mean the same thing.
Section 3.3
Graphing
Equations with
Slope
Okay, now that you know how to graph a line by getting some points,
and you know how to find the slope between two points, you should be able to
find the slope of a line once you have an equation:
Example: Find the slope and graph the line
3x-4y=2
Well, if we find a couple of points: (2,1) and
(6,4), the graph must look like this:
(6,4)
(2,1)
Then finding the slope, we can just use the
same method that we have done the other ones
we get the slope m = 64−−12 = 34 .
Trying this a couple of times on various equations, you might get something
like the following:
Equation
3x – 5y = 10
Slope:
m= 53
2x + 9y = 4
m = - 92
m = -5
m = 13
5x + y = 15
x-3y = 12
I hope that you kind of see a pattern emerging that you
would be able to use as a shortcut. Do you see how the
change in y is always the coefficient of x? And do you see
that the change in x is always the opposite of the coefficient
of y?
These equations are all written the same way and
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have the same pattern for getting the slope without actually figuring it out from
a couple of points.
Here is a pattern for another common way of writing lines. Pick out the pattern
here:
Equation:
Slope:
y = -2x – 5
m = -2
y = 73 x + 4
m = 73
The pattern here is even easier than the first one. When y is
y = - 94 x – 13 m = - 94
by itself, the slope is simply the number in front of x. No
x
1
y= 7-2
m= 7
change.
Since these are two very common ways of writing lines they need some
comparison.
Standard Form:
Slope-intercept Form:
- It is written in the form Ax+By = C where
- Written in the form y = mx+b.
A, B, and C are integers (usually).
- m is the slope without any adjustment.
- Intercepts are found by putting in 0 for
- (0,b) is the y-intercept.
either x or y so each is relatively easily
found as (0, CB ) and ( CA ,0).
-
Slope is always m = - BA
Advantages:
It has no fractions.
Both x- and y- intercept have same amount of
calculation
Disadvantages:
Minor calculation for y-intercept.
Remembering to put the negative sign on the
slope.
Advantages:
Slope is most easily found.
Y-intercept is most easily found.
Prepares you for function f(x) notation.
Disadvantages:
Fractions are often part of the equation.
Since both of them will be given to you to graph, you should be able to work
with both of them.
Important Note: You should also see that we can change from Standard form
into Slope-intercept form (and vice-versa) simply by solving for y. In the
example 3x-4y=2, we get:
3x-4y=2
-4y=-3x+2
Standard: 3x-4y=2 Slope-intercept: y= 34 x - 12
y= 34 x - 12
Slope: m= 34
Slope: m= 34
y-intercept: (0, - 12 )
y-intercept: (0, - 12 )
x-intercept: ( 23 ,0)
x-intercept: ( 23 ,0)
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They are simply two different ways to write the same equation. There is no
difference, except that of convenience. The first way is called standard form,
and the second is called slope-intercept form. Again, they are the same line!
Every point that works in one will work in the other.
In any case, you will learn and have practice with both forms. Being
able to pick out intercepts and slope from lines will help you to graph them
quickly. Having the slope especially makes it a cinch to graph lines. You only
need to find one point, then follow the slope to the next point and draw the
line.
Example:
Graph the line and find the slope of y=- 53 x - 4
Well the slope is right in front of x, so m= - 53
One easy point is to stick in zero for x. We get
the point (0,-4). Following the slope, (it is
negative, so we will head down as we go to the
right) down 5 over 3 and we come to the point
(3,-9), and then draw the line.
-5
7
2
3
Another example:
Graph the line and find the slope of 2x-7y=4
Well the slope is the opposite of 2 over 7, so m=- −27 = 72
It appears that the easiest point in this one is the xintercept, so we stick in zero for y and get x=2: (2,0).
Following the slope we move up 2 and over 7 to the
next point (9,2), and then draw the line.
That covers graphing and finding the slope for the vast
majority of equations. As you will recall, there were a couple of special cases
where either the x or the y were missing. We now look to find the slope of
these.
We will work two examples of this:
First: y= -2
Remember how to find a couple of points
that work: (3,-2) and (-1,-2). It gives us
the graph of a horizontal line where y is
always -2:
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Putting those two points in to the formula for finding slope, we get:
− 2 − −2
0
m=
=
=0
−1− 3
−4
which means that all horizontal lines will have a slope of 0.
Second: x=5
Remember how to find a couple of points that
work: (5,2) and (5,6). It gives us the graph of a
vertical line where x is always 5:
Now if we put the points in the slope formula, we
get:
6−2
4
= = bad news. (Division
5−5
0
by zero is undefined.)
which means that all vertical lines have undefined
slope.
m = undefined
m = 15
m=2
To get a feel for slope a little bit better, we are going to take
a little time to look at some slopes. You will notice that
the higher the number, the steeper it is. Common
m =1
sense from that will tell you that a slope of 0 will
belong to a line that is completely flat. Also, you
should see that since numbers get bigger as the
slope gets steeper, the slope of a vertical line
m = 12
would have to be far greater than a billion. On
the other hand, numbers get increasingly
m = 13
large in the negative direction for lines that
are heading down ever steeper. That means that
vertical lines would have to have a slope that
is less than negative one billion.
m=0
Hmmmmmm…. greater than a billion and less
than negative a billion at the same time.
No wonder that division by zero can’t be
m = − 15
done and is undefined.
m = − 23
A word of caution: Since the term “no slope” is
interpreted by some to mean zero slope and by
others to mean that the slope doesn’t exist, we will
m = −1
simply avoid the term. A vertical line has undefined
m = −2
slope and a horizontal line has a slope of zero.
m = −8
m = undefined
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Section 3.4
Writing equations
Now that we can go from the equation of a line to the finding of points, getting
the slope and graphing the line, we are going to work on how to go backwards.
It really isn’t as difficult as it seems. Since from an equation we can get the
slope, we can certainly write an equation from the slope.
Example:
Write an equation of the line that has slope m= 53 , and goes through the point
(5,-2).
There are two ways to do this. Remember with standard form, the slope is the
negative of the first number over the second. In slope-intercept, the slope is
right in front of the x when y is by itself.
Again, please note
that if you take the
standard form and
solve for y, you will
get the slope-intercept
form.
Section 3.5
Parallel and
Perpendicular
Standard
Since the slope is 53 , we know that an
equation would have to be:
3x – 5y = something but what? Ahhh,
here is where we use the fact that
(5,2) has to work in the equation. If
we stick in 5 for x and 2 for y we get:
3(5) – 5(2) = 15 – 10 =5
3x – 5y = 5
Slope-Intercept
Since the slope is 53 , we know that an
equation would have to be:
y = 53 x+b but what is b? Ahhh, here
is where we use the fact that (5,2) has
to work in the equation. If we stick in
5 for x and 2 for y we get:
2 = 53 (5) + b
2=3+b
-1 = b
Thus our equation must be
y = 53 x - 1
As a side note on slope:
When two lines have the same slope, or steepness such that they never
cross, we call these parallel. When
a
two lines meet at a 90 degree angle,
it is called perpendicular. Let’s
suppose that line “a” has a slope of
7
7
2 ; it is pretty steep and positive.
Line “b” will have a similar ratio, but -2
2
we can see that it is shallow and
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negative. As we can see from the
picture, the slope of “b” is m= - 72 .
b
This pattern happens every time that
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two lines are perpendicular. As a rule: A perpendicular slope is the negative
reciprocal. Some people like to think of it that the two slopes will always
multiply together to give you -1.
As a special case, can you see what slope would be perpendicular to 0? Since
vertical and horizontal are perpendicular, an undefined slope is the answer.
Here are a few examples:
Section 3.6
Line Inequalities
Equation
Slope
Parallel slope
Perpendicular
3x+2y=7
m= - 32
m= - 32
m= 23
y = 5x-2
m=5
m=5
m= - 15
4x-7y=7
m= 74
m= 74
m= 74
x=-7
m = undefined
m = undefined
m=0
y=3
m=0
m=0
m = undefined
Now that you know everything there is to know about lines, we are going to
change the equation into an inequality. Way back when we were studying
linear equations, like 3x - 2 = 7, we finished off by moving to linear
inequalities like, 3x - 2 > 7.
Now we are going to move from lines like 3x - 4y = 8 to 3x - 4y > 8.
As we recount how we did 3x - 2 >7:
3x > 9
x>3
Then we graphed all the
x>3
x<3
points that worked. Please note and
pay attention to the fact that the
number 3 divides the number line
x=3
into three sections:
Have you ever thought about why we graph the number 3 when it really
isn’t a solution? Even though it doesn’t work, it is the boundary between points
that work and those that don’t. Anyway, the process we went through to get
that graph is crucial:
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1) Treat as though equal
2) Mark the border (open or closed)
3) Shade the right side.
The same exact process applies to line inequalities. The example we are
going to use is
3x - 4y > 8
1) We ignore the inequality long enough to graph the border of those that
work and those that don’t.
2) We just need to color all the points that fit the problem that we were given.
But we also need to note that the line itself (where 3x – 4y = 8) is not part
of the solution. We do that by dotting the line. (This is similar to the open
circle on the number line.)
3x – 4y < 8
3x – 4y = 8
3) When we stick any random points
in for x and y (take a minute and
just try a few from anywhere on
the graph), we see that the line
divides the plane into three regions
as shown. Be sure to shade the
correct side.
3x – 4y > 8
Note: It isn’t always safe to trust the direction of the inequality. It is always safe to test a point
or two to make sure you have the right side.
Assignment #39
1. Three types of bears are in a national park. The number of grizzly bears are 4 more than twice
the number of black bears, and 50 more panda bears than black bears. There are a total of 874
bears in the park. How many of each kind are there?
2. An international phone call costs 35¢ to connect and 12¢ for every minute of the call. How
long can a person talk for $3.60?
3. A 52m rope is cut so that one piece is 18m longer than the other. What are the lengths of the
pieces?
4.
Original Price:$292.50
Discount:20%
Final Price:
Original Price:
Discount: 40%
Final Price: $73.90
5.
Solve and graph.
6. 5(x-2) > 7x +8
Fill out the table for each of the following:
Ex. 1
3x + 4y = 7
3.1
x
2
3(2) + 4y = 7
4y = 1
y = 14
7. x + y = 9
x
5
-4
y
0
1
5
3(0) + 4y = 7
4y = 7
y = 74
y
3
0
7
0
Solution:
3x + 4(0) = 7
3x = 7
x = 73
3x + 4(1) = 7
3x = 3
x=1
x
2
y
7
3
0
1
- 134
0
1
5
1
4
7
4
3x + 4(5) = 7
4x = -13
x = - 134
8. 2x – y = 5
x
2
0
-1
y
0
4
9. 5x + 4y = 9
x
1
0
-3
y
0
5
10. x–7y = 13
x
y
1
3
2
0
-1
Assignment #39
Graph the following lines, and label three points.
Example:
2x – 7y = 3
Pick three numbers to make a table (intercepts are helpful):
x
0
1
-2
. ..
y
(0,- 73 ) (1,- 17 ) (-2,-1)
11. 3x + y = 10
12. y = 2x
13. x – 4y = 7
14. x = 3
15. y = - 73 x + 4
16. 6x – 5y = 12
17.
18. 5x + 2y = 6
y = -4
Preparation
19. After reading a bit of section 3.2, try to find the slope between (4,1) and (7,11).
Assignment #39
Answers:
205 Black, 414 Grizzly,
1.
255 Panda
27 minutes
2.
3.
17m, 35m
4.
5.
6.
$234
$123.17
x < -9
-9
7.
x
5
-4
6
9
2
x
2
0
-1
૞
૛
ૢ
૛
10.
y
-1
-5
-7
0
12. (0,0) (1,2) (2,4)
17. (0,-4) (2,-4) (37,-4)
13. (7,0) (3,-1) (0,ିళర)
18. (0,3) (2,-2) ( లఱ ,0)
14. (3,0) (3,1) (3,2)
19. m = భబయ
4
x
1
0
y
1
-3
6
0
ૢ
૞
૚૚
ି
૞
0
y
4
13
3
0
7
8.
9.
11. (0,10) (3,1) (-1,13) 16. (2,0) (0,-భమఱ) (7,6)
ૢ
૝
5
x y
20 1
34 3
2 ିଵଵ
0
଻
ଵଷ
ି
଻
6
-1
15. (0,4) (7,1) (14,-2)
Assignment #40
1. Three types of trees are in a local park. There were 4 more than twice as many aspens as oaks,
and 50 more maples than oaks. There are a total of 874 trees in the park. How many of each kind
are there?
2. If the length is 3 more than 4 times the width of a rectangle and the perimeter is 76mm, what
are the dimensions?
3. If grading in a class is set up so that 10% is attendance, 10% is tutoring, 30% homework, and
50% tests, what is a student’s grade if he has 80% attendance, 50% tutoring, 50% homework and
80% on tests?
4.
Original Price:$392.50
Discount:20%
Final Price:
Original Price:
Discount: 45%
Final Price: $73.90
5.
6. Four consecutive odd integers add up to 328. What are the four numbers?
3.1
Fill out the table for each of the following:
7. 2x + y = 9
x
5
-4
y
3
0
7
8.
y = 5x+2
x
2
0
-1
y
0
4
9. x + 4y = 9
x
1
0
-3
10. y = యళx - 13
y
0
5
Graph the following lines, and label three points.
11. 3x + 2y = 10
12. y = 2x - 7
13. y = భమx
14. x = -6
15. y = - 73 x - 2
16. 2x – 5y = 12
x
y
2
5
2
0
-1
Assignment #40
17.
y=5
18. 5x + y = 6
Find the slope between each pair of points.
3.2
Ex. (7,2) (-3,5)
5−2
7 − −3
3
=
10
m=
19. (5,-2) (7,3)
20. (4,1) (-5,6)
21. (5,-1) (-3,-8)
22. (7,3) (-2,3)
23. (-5,2) (4,-3)
24. (-6,1) (-6,5)
25. Explain the difference between a slope of zero and an undefined slope.
Preparation
26. Find two points of each line and then use those points to find the slope.
2x – 3y = 1
y = యఱx + 4
Assignment #40
Answers:
205 Oaks, 414 Aspen,
1.
255 Maple
w=7, l=31
2.
3.
68%
4.
5.
6.
7.
$314
12. (0,-7) (1,-5) (2,-3)
17. (0,5) (-2,5) (3,5)
13. (0,0) (2, 1) (8,4)
18. (0,6) (1,1) ( లఱ ,0)
14. (-6,0) (-6,1) (-6,2)
19.
20.
21.
22.
23.
24.
25.
79, 81, 83, 85
ૢ
૛
1
y
-1
17
3
0
7
8.
x
2
0
-1
- ૛૞
૛
૞
10.
16. (6,0) (0,-భమఱ) (1,-2)
$134.36
x
5
-4
3
9.
11. (0,5) (భబ
,0) (2,2)
య
y
12
2
-3
0
4
x
y
1
2
ૢ
0
૝
-3
3
0
9
-11 5
x y
35 2
42 5
2 ି଼ହ
଻
0 -13
28 -1
15. (0,-2) (7,-5) (-7,1)
m = ఱమ
m = - ఱవ
m = ళఴ
m=0
m = -ఱవ
m = undefined
Undefined is straight up and
down, vertical.
0 is horizontal, straight across
26. In class.
Assignment #41
1. Three types of horses are in a local ranch. There are 8 more than twice as many Arabians as
Quarter-horses, and 50 more Clydesdales than Quarter-horses. There are a total of 282 horses at
the ranch. How many of each kind are there?
2. What is the radius of a cone that has Surface Area of 219.91 in2 and a slant height of 9 in?
3. Solve and graph the solution.
3x – 1 > x + 9
4.
3.1
Original Price:$392.50
Tax: 6%
Final Price:
5.
Original Price:
Tax: 7%
Final Price: $73.90
Fill out the table for each of the following:
6. 2x - 3y = 9
x
5
-4
y
3
0
7
7.
y = x+2
x
2
0
-1
y
0
4
Graph the following lines, and label x and y intercepts.
8.
5x + 2y = 10
11. x = 10
9. y = x - 6
10. y = x
12. y = - 73 x +4
13. 7x – y = 14
Find the slope between each pair of points.
3.2
14. (8,-2) (7,3)
15. (8,1) (-5,6)
16. (-3,-1) (-3,-8)
17. (7,9) (-2,3)
18. (-5,2) (4,6)
19. (-6,1) (6,1)
Assignment #41
Graph the following lines giving one point and the slope.
3.3
Ex.
2x – 7y = 3
Find one point: ( 32 ,0) and the slope: m = 72 .
Then graph the point. Then go up 2 and over 7 for the next one:
21. y = 4x + 3
22. y = x - 4
23. x = -6
24. y = - 73 x - 2
25. 3x – 4y = 12
26.
27. x + 4y = 9
28. y = 7
20.
-6x + y = 10
5x + 3y = 10
Preparation
29. Write down 5 equations of lines that have the slope:
m= 7
.
2
Assignment #41
Answers:
56 Quarter-horses, 106
1.
10.
Clydesdales, 120 Arabian
radius = 5 in
2.
3.
x > 20
4.
5.
6.
$416.05
0
$69.07
x
5
-4
9
y
x
2
0
-1
- y
9
2
- 0
4
(0,0) (3,8)
20. (0,10); m = 6
11. (10,0) no y-int
21. (0,3); m = 4
12. (0,4) ( ,0)
22. (0,-4); m = 23. (-6,0); m = undefined
20
3
0
15 7
7.
8.
(0,5) (2,0)
13. (2,0) (0,-14)
9.
(0,-6) ( ,0)
14.
15.
16.
17.
18.
19.
m = -5
m = - m = undefined
m = m = m=0
24. (0,-2); m = - Assignment #41
25.
(4,0); m = 27. (9,0); m = - 26.
(2,0); m = - 28. (15,7); m = 0
29. In class.
Assignment #42
3.1
Fill out the table for each of the following:
1. 2x - 5y = 11
x
5
-4
y
3
0
7
2.
y = x+6
x
2
0
-1
y
0
4
Graph the following lines, and label x and y intercepts.
3.
4x - 2y = 10
4. y = - x - 6
5. y = 5x
Find the slope between each pair of points.
3.2
6. (3,-2) (7,3)
7. (9,1) (-7,6)
8. (5,-1) (-3,-8)
9. (-2,9) (-2,3)
10. (-5,2) (5,6)
11. (19,1) (6,1)
12. Explain the difference between a slope of zero and an undefined slope.
Graph the following lines giving one point and the slope.
3.3
13.
-3x + 4y = 10
16. y = 17
14. y = 2x - 7
15. y = x - 4
17. y = - 73 x - 2
18. 2x – 6y = 12
Write the equations of the lines with the slopes and points:
3.4
Ex.
Write an equation of the line that has slope m = 74 , and goes through the point (2,1). Put
the answer in Standard Form.
From the slope m = 74 , I know that the equation must look like:
4x – 7y = something, so I put in the point to see what it is.
4(2) – 7(1) = 1.
Thus the answer is 4x – 7y = 1.
Assignment #42
Ex.
Write an equation of the line that has slope m = 74 , and goes through the point (2,1). Put
the answer in Slope-Intercept Form.
From the slope m = 74 , I know that the equation must look like:
y=
1=
4
7
4
7
x+b
Put the point in to see what b is.
(2) + b
8
7
1
7
1- = b
- =b
Thus the answer is y =
4
7
x-
1
7
.
19. Write an equation of the line that has slope m= -3, and goes through the point (-4,6). Put the
answer in Standard Form.
20. Write an equation of the line that has slope m= 85 , and goes through the point (3,6). Put the
answer in Standard Form.
21. Write an equation of the line that has slope m=- 23 , and goes through the point (1,-3). Put the
answer in Slope-Intercept Form.
22. Write an equation of the line that has slope m=- 45 , and goes through the point (5,-3). Put the
answer in Slope-Intercept Form.
23. Write an equation of the line that has slope m= 2, and goes through the point (0,5). Put the
answer in Slope-Intercept Form.
24. Write an equation of the line that has slope m=- 17 , and goes through the point (-4,7). Put the
answer in Standard Form.
Preparation
25. What are the similarities between the graphs of the following lines?
2x – 3y = 3 and 3x + 2y = 2
y = 2x – 5
and y = 2x + 7
Assignment #42
Answers:
1.
x y
5
-4 - 13 3
0
23 7
6.
7.
8.
9.
10.
11.
12.
m = m = - m = m = undefined
m = m=0
13.
(0,) m = 14.
(0,-7) m = 2
15.
(0,-4) m = 16.
(0,17) m = 0
17.
(0,-2) m = - Undefined is vertical
0 is horizontal
2.
3.
4.
5.
x
2
0
-1
- 0
- 4
18. (6,0) m = y
13
6
(0,-5) (,0)
,0)
(0,-6) (-
(0,0) (2,10)
19.
20.
21.
22.
23.
24.
25.
3x + y = 6
5x – 8y = -33
y = - x - y = - x + 1
y = 2x + 5
x + 7y = 45
In class