You can put this solution on YOUR website!

Start with the given equation



Subtract 5 from both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide

--------------------------------------------------------------
Answer:
So our answer is

You can put this solution on YOUR website!


Jump to problem # 2
# 1

Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation



---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Combine like terms on the left side


Subtract 45 from both sides


Combine like terms on the right side





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point

---

Jump to problem # 1
# 2

Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Add to both sides


Rearrange the equation


---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute the negative


Combine like terms on the left side


Subtract 4 from both sides


Combine like terms on the right side


Divide both sides by -2 to isolate x



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point

You can put this solution on YOUR website!

Start with the given equation



Distribute


Combine like terms on the left side


Subtract 4 from both sides


Subtract 6x from both sides


Combine like terms on the left side


Divide both sides by -11 to isolate x



Reduce

--------------------------------------------------------------
Answer:
So our answer is (which is approximately in decimal form)

You can put this solution on YOUR website!

Start with the given equation



Distribute


Combine like terms on the left side


Subtract 4 from both sides


Subtract 6x from both sides


Combine like terms on the left side


Divide both sides by -11 to isolate x



Reduce

--------------------------------------------------------------
Answer:
So our answer is (which is approximately in decimal form)

You can put this solution on YOUR website!
Start with the given expression

Factor to get

Factor to get

Factor to get

Factor to get



Combine the fractions


Cancel like terms


Simplify


So simplifies to

You can put this solution on YOUR website!


f(1)=-3, f(2)=-7, f(4)=9, f(8)=305, f(16)=3,297


Graph:


----------------------------------------------------


:


f(1)=2.71828, f(2)=7.38906, f(4)=54.59815, f(8)=2980.9579, f(16)=8886110.5205

Graph:



----------------------------------------------------



f(x) = 3x - 2 :
f(1)=1, f(2)=4, f(4)=10, f(8)=22, f(16)=46


Graph:


----------------------------------------------------





f(x)=log(x):

f(1)=0, f(2)=0.30103, f(4)=0.60206, f(8)=0.90309, f(16)=1.20412


Graph:


----------------------------------------------------

f(x) = x^2 - 5x + 6:
f(1)=2, f(2)=0, f(4)=2, f(8)=30, f(16)=182


Graph:


----------------------------------------------------



Rank from fastest growing to slowest growing:

1)
2)
3)
4)
5)

You can put this solution on YOUR website!


f(1)=-3, f(2)=-7, f(4)=9, f(8)=305, f(16)=3,297


Graph:


----------------------------------------------------


:


f(1)=2.71828, f(2)=7.38906, f(4)=54.59815, f(8)=2980.9579, f(16)=8886110.5205

Graph:



----------------------------------------------------



f(x) = 3x - 2 :
f(1)=1, f(2)=4, f(4)=10, f(8)=22, f(16)=46


Graph:


----------------------------------------------------





f(x)=log(x):

f(1)=0, f(2)=0.30103, f(4)=0.60206, f(8)=0.90309, f(16)=1.20412


Graph:


----------------------------------------------------

f(x) = x^2 - 5x + 6:
f(1)=2, f(2)=0, f(4)=2, f(8)=30, f(16)=182


Graph:


----------------------------------------------------



Rank from fastest growing to slowest growing:

1)
2)
3)
4)
5)

You can put this solution on YOUR website!

Start with the given expression


Convert the expression from radical notation to exponent notation. Remember


Rewrite 64 as


Now distribute the exponent Now distribute the outer exponent to each exponent in the parenthesis. Remember

Now multiply the exponents

Reduce

Now convert back to radical notation


Take the square root of 64 to get 8


So

You can put this solution on YOUR website!
Start with the given expression



Convert the expression from radical notation to exponent notation. Remember




Multiply the exponents. Remember


Multiply


Reduce


So


note: after a while you'll notice that when you take the square root of something like , the exponent simply gets cut in half. So , , etc...

You can put this solution on YOUR website!

You can put this solution on YOUR website!

You can put this solution on YOUR website!
This is something you just have to memorize if a calculator is out of the question

You can put this solution on YOUR website!
Let x=# of orchestra seats and y=# of balcony seats

Since the "students sold 650 tickets", this means that . This is our first equation.


Also, since the total cost of x orchestra seats (at $7.50) and y balcony seats (at $3.50) is $4,375.00, this means that . This is our second equation.


Multiply both sides of the second equation by 10. This will make every number a whole number.


Distribute and multiply.






So we now have this system of equations:




Let's use substitution to solve this system.

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Combine like terms on the left side


Subtract 22750 from both sides


Combine like terms on the right side


Divide both sides by 40 to isolate x



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and



This means that the students sold 525 orchestra seats and 125 balcony seats

You can put this solution on YOUR website!

Let's graph the first equation . To do this, we must first solve for y



Start with the first equation


Subtract from both sides


Rearrange the equation




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -3 and the run is 1. This means that to go from point to point, we can go down 3 and over 1



So starting at , go down 3 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and


---------------------------------------


Let's graph the second equation . To do this, we must first solve for y


Start with the second equation


Subtract from both sides


Rearrange the equation





Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -1 and the run is 1. This means that to go from point to point, we can go down 1 and over 1



So starting at , go down 1 unit


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and



------------------------------


Graph of (red) and (green)


So from the graph, we can see that the two lines intersect at the point (-1,-3).

You can put this solution on YOUR website!
Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients


So let's list the factors of -3 (the last coefficient):



Now let's list the factors of 2 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct rational zeros of the function that could occur

You can put this solution on YOUR website!
note: I added the solution to #2


# 1
Jump to problem #2
Once again, I'll do the first one to get you started (hopefully in the right direction)


Let's graph the first equation . To do this, we must first solve for y


Start with the first equation


Add to both sides


Rearrange the equation





Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is 1 and the run is 1. This means that to go from point to point, we can go up 1 and over 1



So starting at , go up 1 unit


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and



-----------------------------------------



Now let's graph the second equation . To do this, we must first solve for y




Start with the second equation


Subtract from both sides


Rearrange the equation




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -1 and the run is 1. This means that to go from point to point, we can go down 1 and over 1



So starting at , go down 1 unit


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and


------------------------------


Now let's graph the two equations together


Graph of (red) and (green)


So from the graph, we can see that the two lines intersect at the point (2,1).

---

# 2

Jump to problem #1

Let's graph the first equation . To do this, we must first solve for y



Start with the first equation


Subtract from both sides


Rearrange the equation




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -3 and the run is 1. This means that to go from point to point, we can go down 3 and over 1



So starting at , go down 3 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and


---------------------------------------


Let's graph the second equation . To do this, we must first solve for y


Start with the second equation


Subtract from both sides


Rearrange the equation





Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -1 and the run is 1. This means that to go from point to point, we can go down 1 and over 1



So starting at , go down 1 unit


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and



------------------------------


Graph of (red) and (green)


So from the graph, we can see that the two lines intersect at the point (-1,-3).

You can put this solution on YOUR website!
Let's graph the equation




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is 3 and the run is 1. This means that to go from point to point, we can go up 3 and over 1



So starting at , go up 3 units


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and



So we can see that the graph is a diagonal line.

You can put this solution on YOUR website!
The point is (6,5)

You can put this solution on YOUR website!
Start with the given expression


Expand. Remember something like


Now let's FOIL the expression



Remember, when you FOIL an expression, you follow this procedure:


Multiply the First terms:


Multiply the Outer terms:


Multiply the Inner terms:


Multiply the Last terms:


Now collect every term to make a single expression



Now combine like terms


---------------------
Answer:
So foils and simplifies to

In other words,

You can put this solution on YOUR website!
Start with the given expression


Simplify to get


Multiply.


So simplifies to

---

Start with the given expression


Simplify to get


Multiply.


So simplifies to

You can put this solution on YOUR website!
Do you want to graph?



Start with the given equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is 1 and the run is 1. This means that to go from point to point, we can go up 1 and over 1



So starting at , go up 1 unit


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and

You can put this solution on YOUR website!

Start with the given equation

Let's find the x-intercept

To find the x-intercept, let y=0 and solve for x:
Plug in

Simplify

Divide both sides by 3


Reduce



So the x-intercept is (note: the x-intercept will always have a y-coordinate equal to zero)



------------------

Start with the given equation

Now let's find the y-intercept

To find the y-intercept, let x=0 and solve for y:
Plug in

Simplify

Divide both sides by -2



Reduce



So the y-intercept is (note: the y-intercept will always have a x-coordinate equal to zero)

------------------------------------------

So we have these intercepts:
x-intercept:

y-intercept:

You can put this solution on YOUR website!

Start with the distance formula.


Plug in , , , and .


Combine like terms


Square each term


Multiply


Add


Take the square root of 256 to get 16


So our answer is


So the distance between the two points is 16 units.

You can put this solution on YOUR website!
Start with the given equation


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 32 and it's opposite -32




Now lets focus on the first equation


Divide both sides by -4 to isolate x



Divide

---

Now lets focus on the second equation



Divide both sides by -4 to isolate x



Divide





So the solutions to are:

or

You can put this solution on YOUR website!

First add the x-coordinates together to get


Divide that result by 2


So the x-coordinate of the midpoint is


Now add the y-coordinates together to get


Divide that result by 2


So the y-coordinate of the midpoint is


So the midpoint between the points and is

You can put this solution on YOUR website!
you need to post the problem

You can put this solution on YOUR website!
a) You are correct.

b) To find the fair price, replace the 2 with x and the -1.14 with 0 (since a "fair" price means that the player will break even) to get




Multiply


Combine like terms


Subtract from both sides


Multiply both sides by to completely isolate x



Multiply


Use a calculator to approximate the fraction


Round to the nearest hundredth


So the fair price of the ticket is about 86 cents.

You can put this solution on YOUR website!
Expected Value:




So he expects to earn $0.40

However, since they paid $2, this means that

0.40-2.00=-1.60


So they really expect to lost $1.60


So the correct answer is C)

You can put this solution on YOUR website!
I'm assuming that order does not matter.





Since order does not matter, we must use the combination formula:


Start with the given formula



Plug in and



Subtract to get 12


Expand 19!



Expand 12!




Cancel



Simplify


Expand 7!




Multiply 19*18*17*16*15*14*13 to get 253,955,520



Multiply 7*6*5*4*3*2*1 to get 5,040



Now divide



So 19 choose 7 (where order doesn't matter) yields 50,388 unique combinations

You can put this solution on YOUR website!
Start with the given expression


Since the denominators are equal, this means that we can combine the fractions


Distribute


Combine like terms

Reduce


So

You can put this solution on YOUR website!
First FOIL

Start with the given expression


Expand. Remember something like


Now let's FOIL the expression



Remember, when you FOIL an expression, you follow this procedure:


Multiply the First terms:


Multiply the Outer terms:


Multiply the Inner terms:


Multiply the Last terms:


Now collect every term to make a single expression



Now combine like terms


So foils and simplifies to

In other words,

---

Now let's FOIL



Expand. Remember something like


Now let's FOIL the expression



Remember, when you FOIL an expression, you follow this procedure:


Multiply the First terms:


Multiply the Outer terms:


Multiply the Inner terms:


Multiply the Last terms:


Now collect every term to make a single expression



Now combine like terms


So foils and simplifies to

In other words,

---

So now becomes


Start with the given expression


Combine like terms



So