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Remember,

Profit = Revenue - Cost

which is symbolically:





Plug in and


Distribute the negative.


Combine like terms


Now the break even point occurs when you neither gain money nor lose money. In other words, break even point happens when the profit is zero. So this means that the break even point occurs when


Plug in


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


or Break up the expression.


So the answers are or


which approximate to or


So the break even points occur when the price is either $4.33 or $11.67

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Vertex:


In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: .


Start with the given formula.


From , we can see that , , and .


Plug in and .


Negate to get .


Multiply 2 and to get .


So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also .


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the y-coordinate of the vertex is .


So the vertex is which in decimal form is (3.5, -0.25).


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


x-intercept(s):


Start with the given equation.


Plug in y=0



Start with the given equation


Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
or

or Now solve for x in each case


So the answers are

or


which means that the x-intercepts are (4,0) and (3,0)


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

y-intercept:


Start with the given equation.


Plug in x=0


Simplify


So the y-intercept is (0,12)



So with all of this info, we get this sketch:

Graph of


note: the point (7,12) is a reflection of the point (0,12)

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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 10 and the run is 1. This means that to go from point to point, we can go up 10 and over 1



So starting at , go up 10 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

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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 10 and the run is 1. This means that to go from point to point, we can go up 10 and over 1



So starting at , go up 10 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

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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 10 and the run is 1. This means that to go from point to point, we can go up 10 and over 1



So starting at , go up 10 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

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Are you sure that the expression is not ???


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,3,5,6,10,15,25,30,50,75,150
-1,-2,-3,-5,-6,-10,-15,-25,-30,-50,-75,-150


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*150
2*75
3*50
5*30
6*25
10*15
(-1)*(-150)
(-2)*(-75)
(-3)*(-50)
(-5)*(-30)
(-6)*(-25)
(-10)*(-15)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	150	1+150=151
2	75	2+75=77
3	50	3+50=53
5	30	5+30=35
6	25	6+25=31
10	15	10+15=25
-1	-150	-1+(-150)=-151
-2	-75	-2+(-75)=-77
-3	-50	-3+(-50)=-53
-5	-30	-5+(-30)=-35
-6	-25	-6+(-25)=-31
-10	-15	-10+(-15)=-25

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term

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


Answer:


So factors to .


Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term


Condense the terms

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


Answer:


So factors to .


Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

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The sentence "Four more than the quotient of a number and 3 is at least that number" translates to




Start with the given inequality.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Divide both sides by to isolate . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


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

Answer:

So the answer is

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The sentence "The sum of a number and 81 is greater than the product of 3 and that number" translates to




Start with the given inequality.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Divide both sides by to isolate . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


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

Answer:

So the answer is


Which approximates to

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Table of Contents:


Jump to Table
Jump to Graph


In order to graph , we need to plot some points.


Let's find y when
(Note: you can start with any x-value):


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


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





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Now let's make a table of the values we just found.

Table of Values:

x
y
-9
0
-8
9
-7
16
-6
21
-5
24
-4
25
-3
24
-2
21
-1
16
0
9
1
0

Now let's plot the points:







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Graph:

Now draw a curve through all of the points to graph :


Graph of

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I'll do the first two to get you started

The slope of any line in the form of y=k where k is any number is 0 since the equation is a horizontal line. So the slope of is 0

So the graph of is:

Graph of


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


The slope of any line in the form of x=k where k is any number is undefined since the equation is a vertical line. So the slope of is undefined.

So the graph of is:

Graph of

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I'm not too familiar with the TI 84 plus, but this page might be able to help.

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I'll do the first two to get you started:


1)


Start with the given system of equations:




Let's use substitution to solve the 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


Divide both sides by


Break up the fraction


Reduce



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

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


Multiply both sides by the LCM of 2. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



Distribute and multiply the LCM to each side



Combine like terms on the left side


Add 64 to both sides


Combine like terms on the right side


Simplify

Since this equation is never true for any x value, this means there are no solutions.

---

2)





Start with the given system of equations:




Let's use elimination to solve the system


Multiply the both sides of the second equation by 5.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the y terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms


Divide both sides by -5.




So our answer is and .


Which form the ordered pair .

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I'm assuming that the equation should look like this:


Start with the given expression


Group like terms


Factor out the GCF out of the first group. Factor out the GCF out of the second group


Since we have the common term , we can combine like terms


Factor to get


Now set each factor equal to zero:

, or


Now solve for x for each factor:


, or


So the solutions are , or

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Remember,

1 kg = 1000 g or 1 kg = 1*10^3 g

and

1 km = 1000 m = 100,000 cm or 1 km = 1*10^3 m = 1*10^5 cm


So 1,200,000,000 kg = 1,200,000,000,000 g or 1,200,000,000 kg = 1.2*10^12 g


and 1 km^3 = (100,000)*(100,000)*(100,000) cm^3 = 1,000,000,000,000,000 cm^3 = 1*10^15 cm^3


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

So

1,200,000,000 kg = 1.2*10^12 g

and

1 km^3 = 1*10^15 cm^3


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

Also, remember that

 
            Mass
Density =  ---------
            Volume

which is symbolically






Start with the given equation.



Plug in the given mass M=1.2*10^12 g and the given volume V=1*10^15 cm^3


Divide 1.2 by 1 to get 1.2. Divide by by subtracting


Subtract the exponents.


So the density is g/cm^3

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First let's find the slope of the line through the points and


Start with the slope formula.


Plug in , , , , ,


Subtract from to get


Subtract from to get


Reduce


So the slope of the line that goes through the points and is


Now let's use the point slope formula:


Start with the point slope formula


Plug in , , and


Distribute


Multiply


Add 4 to both sides.


Combine like terms.


Simplify


So the equation that goes through the points and is


Notice how the graph of goes through the points and . So this visually verifies our answer.
Graph of through the points and

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Start with the given expression.


Distribute.


Multiply


So

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Start with the given expression.


Expand.


Distribute.


Multiply.


Now combine like terms.


So expands to .


In other words, .

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Start with the given expression.


Multiply the 1st term by


Combine the fractions.



FOIL


Multiply the 2nd term by



Combine the fractions.


Subtract the fractions.


Combine like terms


Distribute


So where or

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So where

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Start with the given system of equations:



Multiply the both sides of the first equation by -2.


Distribute and multiply.


Multiply the both sides of the second equation by 3.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the x terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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Start with the given expression.


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where , , , or

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To find the y-intercept, plug in x=0 and simplify. To find the x-intercept(s), simply use the quadratic formula. Post a problem so I can walk you through it.

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Let x=# of $2 tickets and y=# of $3 tickets


First translate the word problem into the system of equations:




Let's use substitution to solve the 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 1020 from both sides


Combine like terms on the right side


Divide both sides by -1 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 means that 210 $2 tickets and 130 $3 tickets were sold

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So where or

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So where

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note:

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Since you cannot take the square root of -1 and get a real number, by convention where "i" is an imaginary number

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Let x=# of adults and y=# of students
Since the "63 tickets were sold", this means that the first equation is
Also, because the tickets "were sold at $6 for adults and $4.50 for students" and they collected a "total of $358.50", this means that the second equation is

Multiply every term in the second equation by 10 to move the decimal point one spot to the right

So we have the 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 2835 from both sides


Combine like terms on the right side


Divide both sides by 15 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


So there were 50 adults and 13 students

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note: or