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Start with the given equation


Plug in and


Divide both sides by to isolate "c"


Rearrange the equation


Start with the given equation


Plug in and


Plug in


Multiply



Reduce


Multiply both sides by "a"


So the solution is

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The last solution was perfect, but you can simplify the equation:


Start with the given equation


Multiply the second term by


Combine the terms.


Distribute


Add the fractions.


Combine like terms.


So simplifies to

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Is the equation


Start with the given equation.


Take the square root of both sides. Note: don't forget the "plus/minus"


or Break up the "plus/minus" to form two equations.


or Take the square root of to get .


or Add to both sides.


or Combine like terms.


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


Answer:


So the solutions are or .

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Start with the given equation.


Multiply both sides by 100 to clear out the decimals.


Distribute and multiply.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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You're on the right track. Once you have the solution , simply graph by drawing a vertical line through like this


Graph of

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Start with the given equation.


Distribute.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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Start with the given equation.


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.


or Reduce


So the answers are or which are both complex (ie they are NOT real solutions)


which approximate to or

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"consecutive even numbers" are integers such as 2, 4, 6, 8, 10, etc. Algebraically, they follow the pattern , , , etc

So if "The sum of three consecutive even numbers is 150", then the equation is





Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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


Go back to the expression that represents the first number


Plug in


Multiply. So the first number is 48


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


Go back to the expression that represents the second number


Plug in


Multiply.


Add. So the second number is 50



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


Go back to the expression that represents the third number


Plug in


Multiply.


Add. So the third number is 52


==============================

Answer:

So the three consecutive even numbers is 48, 50, and 52


Check:

1)
they are consecutive even integers...check

2)
48+50+52=150
150=150 ... works

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Well, you're on the right track. If , then . On the other hand, if , then . Consider if , then

So breaks down into (if ) and (if )


To graph , first graph and together on the same coordinate axis:

Graph of (red) and (green)


Since we specified that is only true if and is only true if , this means that we only draw the portion that is ABOVE the point of intersection.


So this means that we get:


Graph of

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Since this is not an equation, you can't solve it. However, you can simplify.


Start with the given expression.


Combine the fractions.


Multiply and to get


Multiply and to get


Factor into . Factor into


Highlight the common terms.


Cancel out the common terms.


Simplify


So simplifies to


In other words, where

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I'll do the first two to help you along

# 1



Start with the given equation.


Distribute.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

---

# 2



Start with the given equation.


Distribute.


Combine like terms on the left side.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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I'll do the first one to get you started

# 1



Start with the given equation.


Distribute.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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


Combine like terms.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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


Combine like terms.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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


Combine like terms.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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


Factor to get


Multiply both sides by the LCD to clear out the fractions.


Distribute and multiply


Combine like terms on the left side.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


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

Answer:

So the answer is

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So where every variable is positive.

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So

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Start with the given equation.


Combine like terms.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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Start with the given equation


Subtract from both sides


Combine like terms.


Divide both sides by -1 to isolate "y"


Break up the fraction.


Reduce


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 2 and the run is 1. This means that to go from point to point, we can go up 2 and over 1



So starting at , go up 2 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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Note: this is a system of linear equations problem since we have a system two linear equations with two variables.



Let W=# of wins and L=# of losses

Since they played a total of 162 games (and each game was either a win or a loss), this means that we have the first equation

Also, since the "White Sox won 27 fewer than twice as many games as they lost", this tells us that the second equation is


Start with the first equation


Plug in . In other words, replace "W" with


Combine like terms.


Add 27 to both sides


Divide both sides by 3 to isolate "L"


So the White Sox lost 63 games


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


Go back to the second equation


Plug in


Multiply


Subtract


So the White Sox won 99 games

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Consecutive odd integers are integers that are odd and are next to one another. So numbers like 1,3,5,7,etc are consecutive odd integers. Algebraically, they follow the form , , , etc. where "x" is any integer.


Also, because "six more than nine times the first integer is five times the second integer", this translates to where is the first number and is the second number





Start with the given equation.


Distribute.


Combine like terms on the left side.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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


Now move onto the expression for the first number


Plug in .


Multiply and to get .


Combine like terms.


So the first number is 1.



Now move onto the expression for the second number


Plug in .


Multiply and to get .


Combine like terms.


So the second number is 3.


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

Answer:


So the two numbers are 1 and 3


Check:







works.

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Start with the given equation


Subtract 15 from both sides


Square both sides


Subtract 2 from both sides


So the possible solution is . However, we need to verify it.


Check:


Start with the given equation


Plug in


Add


Take the square root of 16 to get 4


Add


Since the equation is NOT true, this means that is not a real solution.

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

Answer:

So there are no solutions.

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Start with the given equation


Subtract from both sides


Rearrange the terms.

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

So let's complete the square for


Start with the given expression.


Factor out the coefficient . This step is very important: the coefficient MUST be equal to 1.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract inside the parenthesis. Make sure to place this after the "x" term. Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


Combine like terms.


Distribute.


Multiply.


So after completing the square, transforms to . So .


So is equivalent to .


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


Start with the given equation.


Subtract from both sides.


Combine like terms.


Divide both sides by .


Reduce.


Take the square root of both sides. Note: remember the "plus/minus"


or Break up the "plus/minus" to form two equations.


or Add to both sides.


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


Answer:


So the solutions are or .

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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 answer is

(with a multiplicity of 2)

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4) Correct.



5)
a) With Replacement

                   # of red marbles      12
P(First is red) = ------------------ =  ----
                  total # of marbles     25

Since you are replacing the marble (ie putting it back), this means that the total number of marbles stays the same

                      # of black marbles      13
P(Second is black) = ------------------ =  -------
                     total # of marbles       25

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

b) Without Replacement

                   # of red marbles      12
P(First is red) = ------------------ =  ----
                  total # of marbles     25

Now since you are NOT replacing the marble, this means that the total number of marbles will go from 25 to 24

                      # of black marbles            13         13
P(Second is black) = ----------------------    = --------- =  -----
                     total # of marbles - 1         25-1       24

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I'll do the first two to get you started

# 1

Start with the given expression


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


Distribute


Combine like terms.



So simplifies to


In other words, where

---

# 2

Start with the given expression


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



So simplifies to


In other words, where

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Well notice how if , then which is NOT defined. Also, if , then which is also NOT defined. So that's why there are domain restrictions of and

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Are you trying to graph ?? If you are, I'm not sure why you are solving for "x" (when graphing you ALWAYS solve for "y")




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 5 and the run is 1. This means that to go from point to point, we can go up 5 and over 1



So starting at , go up 5 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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I'll do the first two to get you started

# 1

Start with the given equation


Subtract from both sides


Subtract from both sides


Combine like terms.


Divide both sides by to solve for

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

Answer:

So the solution is

---

# 2




Start with the given equation.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Break up the fraction


Reduce

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

Answer:

So the solution is

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Jump to transformations of the logarithmic function

# 1

First let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (-2,0.135)


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


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (-1,0.368)


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


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (0,1)


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


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (1,2.718)


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


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (2,7.389)



Now let's plot these points




Now draw a line through the points to graph



================================================


# 2

Now let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (-2,0.018)


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


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (-1,0.135)


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


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (0,1)


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


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (1,7.389)


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


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (2,54.598)


Now let's plot these points




Now draw a line through the points to graph





Notice how the graph of is much steeper than the graph of . This is because the values of are simply the values of squared (since ). For instance, if , then and . Since is the square of , this verifies our claim. So the transformation that occurs on is a horizontal compression (since we are dealing with "x" and the graph gets thinner).

======================================================



Jump to transformations of the exponential function
# 3


Now let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (1,0)

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

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (2,0.301)

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

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (3,0.477)

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

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (4,0.602)

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

Let's find the y value when


Start with the given equation


Plug in


Use a calculator to evaluate to get


So if , then . So we have the point (5,0.699)



Now let's plot the points




Now connect the points to graph



=============================================================================

# 4


Now let's calculate the values for


Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (1,0.301)

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

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (2,0.602)

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

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (3,0.778)

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

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (4,0.903)

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

Let's find the y value when


Start with the given equation


Plug in


Multiply and to get


Use a calculator to evaluate to get


So if , then . So we have the point (5,1)



Now let's plot the points




Now draw a curve through the points to graph






By comparing and , we can see that the graph of is simply the graph of shifted up some number of units. So a vertical translation has occured on to get