You can put this solution on YOUR website!
This is an "interesting" problem. There may be another way to solve this, but I used logarithms.
If you are not familiar with logarithms, then you will need to repost your problem because once
I respond it drops off the "easy to find" list that tutors use to answer problems.
.
I first wrote this problem as:
.

.
On the right side, whenever you divide by a fraction you can invert the fraction and use
the inverted form to multiply the numerator. So dividing the numerator by 0ne-eighth is
the same as multiplying the numerator by 8. This means that and
. Substituting 4096 for the right side changes the equation to:
.

.
Take the log of both sides:
.

.
On the left side you can apply the rule that the log of a product is equal to the sum
of the logs of the two factors in the product. Applying this rule leads to:
.

.
Next, applying the exponential rule of logarithms, in each of the logs on the left side
you can bring out the exponent as a multiplier of the log to get:
.

.
Now recognize that log(4), log(128) and log(4096) are just numbers. You can get them from a
scientific calculator. If you take those logs (I used base 10) you will find that they are:
.
log(4) = 0.602059991
log(128) = 2.10720997
log(4096) = 3.612359948
.
Substitute these values into the equation in place of log(4), log(128), and log(4096) and
the equation becomes:
.

.
On the left side do the distributed multiplication and you have:
.

.
Just to be a little more conventional, let's put the constants in front of the variables in
each of the terms on the left side ... just a slight re-arrangement ...
.

.
Notice that we have two terms involving just x as the variable. They can be combine and
when you combine their multipliers ... [-0.602059991 + 2.10720997] ... you reduce the
equation to:
.

.
Then let's get this into conventional quadratic form by putting all the terms on the left side
and having a zero on the right side. Do this by subtracting 3.612359948 from both sides to get:
.

.
And just to make this a little more easy to recognize, let's divide both sides (all the terms)
by the multiplier of the x-squared term. In other words, divide both sides by 0.602059991 and
the equation becomes:
.

.
That sure makes things easier to see ... and easier to work with. Now you have this in a
simpler to work with quadratic equation. In fact, with a little thought, this equation can
be either factored or you can use the quadratic formula to solve for x. The left side factors
into:
.

.
and this makes the equation become:
.

.
This equation will be true if either of the factors on the left side equals zero because a
multiplication by zero on the left side makes the entire left side equal to zero and therefore
equal to the right side. So, one at a time set the two factors equal to zero and you have:
.

.
add 1.5 to both sides and this becomes:
.

.
Then set the other factor equal to zero:
.

.
and subtract 4 from both sides to get:
.

.
Check both answers by returning to the original equation:
.

.
and first replacing x by 1.5 and then by -4 to make sure the equation still balances.
.
When x is replaced by 1.5 you have:
.

.
Don't forget that the right side is 4096 so the equation becomes:
.

.
The exponent of the first factor is (2.25 - 1.5 = 0.75) and this makes the equation become:
.

.
Calculator time ... raise the two factors on the right side to their respective exponents and
you get:
.

.
and when you do the multiplication on the left side you get 4096, so when you use 1.5 for x
it works.
.
Next check when you use -4 for x. The original problem becomes:
.

.
The exponent of the first factor on the left side simplifies to give:
.

.
Add the 16+4 to get an exponent of 20 and note that the negative exponent of the second factor
makes the second factor equivalent to
.
this makes the equation become:
.

.
But 4 to the 20th power is 1.099511628 *10^12 and when this is divided by 128 to the 4th
power (which is 26835456) the result is again 4096. So the original equation reduces to:
.
4096 = 4096
.
and therefore the original equation holds true when x = -4
.
The two answers check. Those answers are x = 1.5 and x = -4
.
There probably is an easier way to do this, but this way works.
.
Hope this helps you to see one way to do the problem.
.

You can put this solution on YOUR website!
First, let T represent the tuition cost that you are going to pay and C represent the credit
hours that you schedule.
.
Then ask yourself how you would compute your tuition based on the number of credit hours that
you sign up for. If you take only a 1 credit hour course, how much would you pay in tuition?
The answer would be $65 would it not? How about if you sign up for only a 3 credit hour course?
You would then have to pay tuition of $195. How about if you are a full time student and
you carry 16 credit hours? How much would you expect to pay in tuition?
.
Maybe you can see from this that you are computing your tuition bill by multiplying
the number of credit hours you plan to take by the $65 fee for each credit hour. In other
words you have a tuition bill of $65 times C where C is the total number of credit hours.
In equation form this is:
.
T = 65*C
.
Since T depends on the number of credit hours, we say that T is a function of C or just
that T is equal to f(C). Therefore in functional notation we can write:
.
f(C) = 65*C
.
Or if you care to identify the left side of this equation as relating to tuition, you
could more appropriately write it as:
.
T(C) = 65*C
.
which makes it a little clearer to remember as being "the tuition as a function of C,
the credit hours, is equal to 65 times the number of credit hours" with the answer to
be expressed in dollars.
.
Hope this helps you to understand the problem a little better.
.

You can put this solution on YOUR website!
Let's start by saying that the selling price is represented by P.
.
Next, if the buyer pays P dollars for the car, what's the first thing that happens? The
broker takes 10% or one-tenth (0.1) of P as the charge for his work in selling the car. Ronald
then gets what's left. So P dollars is the selling price and after the broker takes 0.1P
as the cost of his work, then what's left is P minus 0.1P or 0.9P.
.
Therefore, the money that Ronald has left to pay off the loan is 0.9P. And Ronald needs that
amount to be at least the amount he owes on the car. (If it were more than he still owes,
I'm sure that he'd like that very much because he could spend the rest or save it for a
"rainy day."
.
So writing this as an inequality of requiring 0.9P to be at least equal to or greater than
the amount owed we can say:
.
0.9P ≥ $11025
.
To solve this for P you simply divide both sides of this inequality by the multiplier of
P just as you would in an equation. The multiplier of P is 0.9, so divide both sides of the
inequality by 0.9 as follows:
.
0.9P/0.9 ≥ $11025/0.9
.
On the left side of the inequality the 0.9 in the numerator cancels with the 0.9 in the
denominator and you are left with just P. On the right side when you divide the $11025 in
the numerator by the 0.9 in the denominator, you get $12,250. This means that the problem
is reduced to:
.
P ≥ $12,250
.
This says that Ronald needs to have the broker sell the car for $12,250 or more in order
for there to be enough to give the broker 10% ($1225) for selling the car and to pay off
the $11,025 still owed on the loan. Note that $1225 + $11,025 does equal the selling price
that Ronald at least needs. If the broker sells the car for more that $12,250 what happens?
Say the broker sells it for $13,000. The broker takes 10% or $1,300 and gives the remaining
$11,700 to Ronald, and Ronald pays off his $11,025 loan from that amount. He has $11,700 minus $11,025
which is $675 left to use in whatever way he chooses.
.
Hope this helps you to understand this inequality problem and how you can think your way through
to a solution.
.

You can put this solution on YOUR website!
A couple of minor corrections:
.
Your work said:
.
<=== this is ok
.
f(x)=2/3+2 <=== a couple of corrections here. The x was left out on the right side and the
constant should have a minus sign. The answer should be:
.

.
You can get this into conventional graphing mode by replacing f(x) with y and you have:
.

.
With this equation you can find coordinate points on the graph. For example, let x equal 0. The
corresponding value of y is then determined by substituting 0 for x in the equation for
y. When you do that substitution you find that y = -6. So you then know that the point
(0, -6) is on the graph. Then let x = 3. When you substitute 3 for x in the equation the
right side becomes 2 - 6 = -4. This tells you that the point (3, -4) is on the graph.
And you can continue this process until you get enough points to determine the graph.
.
You should find that the graph looks like this:
.

.
Hope this helps you to understand the problem and how you can work it.
.

You can put this solution on YOUR website!
In this problem you are not going to get a numerical answer for y. What you are going to get
is y by itself on one side of the equation, and everything else on the other side.
.
Start with the given equation:
.

.
To get y by itself on one side of the equation, let's begin by getting rid of the 5x that is
on the same side of the equation as the term that contains the y. Do this by subtracting
5x from both sides of the equation. When you subtract 5x from both sides the 5x on the left
side disappears and a minus 5x shows up on the right side as follows:
.

.
Now you can solve for y by dividing both sides of this equation (all terms) by 6. On the
left side when you divide 6y by 6 you get just y. Then you have to divide all the terms
on the right side by 6 also. This makes the equation become:
.

.
which can also be written as:
.
if you prefer.
.
The right side then simplifies to:
.

.
And that's the answer to the problem. Hope this helps to clear it up for you.
.

You can put this solution on YOUR website!
Let P represent the "period of vibration." And L represent the length. Since the period of
vibration varies directly as the square root of the length, we can write the formula as:
.

.
K is called the "constant of proportionality." It is a constant that multiplies the square
root of L to make the answer equal to P. Since K is a constant, we can tell from the equation
that if L increases, then P must increase also to keep both sides of the equation equal.
.
The first question is, "What is the value of K?"
.
From the information in the problem we know that when P = 3, then L = 36. Substitute these
two values into the equation and you have:
.

.
But the square root of 36 is 6. Replace the square root of 36 by 6 and the equation becomes:
.

.
Solve for K by dividing both sides of this equation by 6 to get:
.

.
And simplifying the fraction we get:
.

.
Now that we have K we can substitute for K in the general equation and we get:
.

.
Now to solve for any period all we have to do is to substitute the Length. For this problem
we are told to find the "period of vibration" when L= 5.0625 inches. Substitute this value of
L into the equation and you have:
.

.
Take the square root of 5.0625 and you get 2.25. Then substitute that value into the equation
in place of and the equation becomes:
.

.
Multiply out the right side and you have:
.

.
So the answer to this problem is the period of vibration is 1.125 seconds when the length
of the arm of the pendulum is 5.0625 inches.
.
Hope this helps you to understand the problem and how to work it.
.

You can put this solution on YOUR website!
Given:
.
"There are 7 students each student has 7 cages. Each cage has 7 cats each cat has 7 kittens. How many legs are there?"
My work and answer as are follows.

student 1 <==== ok
cage 1 <==== ok
7 cats <=== ok
49 kittens <=== ok
224 ft per cage <=== ok. 196 kitten legs (4 times 49) + 28 cat legs (4 times 7)
1568 ft per kid + 2 ft <=== ok. 224 per cage times 7 cages per student = 1568 and add 2 student legs
.
1570 legs per student times 7 students = 10,990 <=== ok
I have the answer as 10,990 legs <=== That's what I got also ...

You can put this solution on YOUR website!
Let t (the number of seconds for 1 complete swing of the pendulum) be represented on the
vertical axis (y-axis). And let L (the length of the pendulum arm in meters) be represented
on the horizontal axis (x-axis)
.
You can easily calculate a couple of points on the graph of:
.

.
just to check what your calculator shows as the graph. First you can tell that L has to be
a positive value. Why? Because if it were negative you would be taking the square root of
a negative number and this means that the answer would not be a real number. (Also it doesn't make
sense that a pendulum would involve a negative length.)
.
What if L is zero? If you substitute zero for L in the equation, the square root term becomes zero
and this makes the whole right side of the equation become zero. So when L = 0 then t also
equals zero. This means that the point (0,0) is on the graph. Note that this is just a
mathematical solution. It really makes no sense to have a pendulum with an arm of zero length.
.
What if L is 9.8? Then the term in the radical becomes 1 and the square root is 1. This
makes the right side of the equation become:
.

.
and is approximately 6.28. So the point (L, t) is (9.8, 6.28) and it should be
on the graph.
.
These are points on the graph just to help us get a feel that our graph is correct.
.
When you do the graph on your calculator it should look like this:
.

.
Note that (0,0) is a point on the graph. And when you go to 9.8 on the horizontal
axis, the corresponding value in the vertical direction appears to be 6.28 just as we had
said. The graph appears to be correct.
.
Now all you have to do to answer the problem is:
.
First, go out to 1.5 on the horizontal axis and determine how many vertical units you have to go
up to get to the graph. (You should get an answer of about 2.458 seconds.)
.
Then go out to 2.5 on the horizontal and determine from that point how many vertical units
you have to go up to get to the graph. (You should get an answer of about 3.173 seconds.)
.
Hope this helps you to understand the problem and how to solve it.
.

You can put this solution on YOUR website!
Let M represent the number of members who donated to the charity. And let C represent the
amount of money each of these members contributed.
.
Given this, then the total amount of money collected can be determined by multiplying
M times C. But the problem tells you that the total amount collected was $2040. Therefore,
we can set these two equal. In equation form this first equation is:
.

.
The problem then tells you that if the number of members is increased by 69 (that would mean
the number of members becomes M + 69) the money collected from each member could be 23 dollars
less (meaning C - 23). The total amount of money collected can then be found by multiplying
the new number of members times the new amount collected from each member and setting
that equal to $2040. In equation form this second equation is:
.

.
We now have two equations involving two unknowns. If a solution exists, this should be
all we need.
.
Returning to the first equation, we can solve for C in terms of M by dividing both sides
of the equation by M to get:
.

.
Then we can substitute the right side of this for C in our second equation as follows:
.

.
You can multiply this out by, one at a time, multiplying each of the terms in the first
set of parentheses times each of the terms in the second set of parentheses and then simplifying
the result by collecting the like terms.
.
Multiplying the M in the first set and each of the two terms in the second set results in:
.

.
Then multiplying the 69 and each of the terms in the second set gives:
.

.
Combining these two products on the left side and remembering that they equal 2040 results
in:
.

.
Notice that you have 2040 as a term on each side of the equation. You can eliminate both
by subtracting 2040 from both sides to get:
.

.
Get rid of the denominator M by multiplying both sides of the equation (all terms) by M.
This makes the equation become:
.

.
Just to make things a little more normal, let's first get rid of the minus sign on the term
with the M squared by multiplying the entire equation (both sides and all terms) by -1.
The equation then becomes:
.

.
Rearrange the terms so that they are in descending powers of M:
.

.
Notice now that 23 is a common factor of all the terms on the left side because 69 equals
3 times 23. So get rid of the factor 23 by dividing both sides of the equation (all terms)
by 23 to reduce the equation to:
.

.
Finally, multiply out the term 3*2040 to get 6120 and this makes the equation become:
.

.
This is now in the conventional form of a quadratic equation. You can solve it by using the
quadratic formula or by factoring. (The quadratic formula is a universal way because it
applies to quadratic equations in general. In this case it is probably easier to apply than
trying to factor this quadratic. However, I was able to get the quadratic factored and
in factored form the equation became:}
.

.
This equation will be true if either of the factors is equal to zero because multiplying
the left side by zero will make the left side become equal to the right side. Therefore,
let's set the two factors (one at a time) equal to zero.
.

.
Solve for M by subtracting M from both sides to get:
.

.
That doesn't make sense ... a minus number of members??? Let's toss out that solution and
proceed by setting the second factor equal to zero:
.

.
Solve for M by adding 51 to both sides to get:
.

.
This looks like a good answer. Let's check it.
.
The club currently has 51 members. To get $2040 each member contributed $2040 divided
by 51 and that is $40.
.
If the club had 69 more members (or a total membership of 51 + 69 = 120) and each of these
members contributed $23 less than the $40 originally contributed (or each contributes
$40 - $23 = $17) then you should still have $2040. Multiply 120 members times $17 each
and the product is $2040. Our answer checks.
.
The answer to this problem is that the club currently has 51 members.
.
Sort of complex to explain. I hope this helps you to understand the problem and is clear enough
that you can see one way to solve it.
.

You can put this solution on YOUR website!
Given:
.

.
Since the denominators of both terms are we can get rid of them by multiplying
both sides of this equation by the quantity . That multiplication leads to:
.

.
Then cancel the terms in the numerators with the same terms in the denominators:
.

.
What you are left with is:
.

.
To solve, take the square root of both sides to get two possible answers:
.
and
.
because if either 2 or -2 is squared, the result is +4.
.
We need to check out these two answers. Let's look first at x = +2. Go to the original
problem and substitute +2 for x. oops. Notice what happens. Both terms have (x - 2) in the
denominator. When x is +2 and you subtract 2 from it, the denominator becomes 0. But division
by zero is not allowed in algebra. Therefore, x cannot equal +2.
.
What about if x = -2. If we substitute -2 for x in the original problem, it becomes:
.

.
(-2 - 2) is equal to -4. So we can substitute -4 for both of the denominators to get:
.

.
Then you can note that squaring -2 results in +4. You can substitute +4 for the numerator on
the left side. When you do that substitution the equation becomes:
.

.
Without going further, you can see that the left side equals the right side. Therefore,
if you take -2 as the value of x, it will maintain the equality of both sides of the equation.
.
Therefore, the answer to your problem is x = -2.
.
Hope this helps you to understand the problem and to see the procedures that you can use to
solve it.
.

You can put this solution on YOUR website!
The answer to this is g ... 2n + 1.
.
Why? Because doubling any number makes it even, and then adding 1 to that even number makes
the term an odd integer.
.
Let's show an example of this. Suppose that n is an even number, say 8. Then 2n is 16 and when
you add 1 to that you get 17, an odd integer. Then suppose that n is an odd integer, say 9.
Then 2n would be 18 and adding 1 to that again makes the result an odd integer.
.
Now lets try to find examples of why the others won't necessarily give you an odd answer. Answer f
is n + 1. Suppose n was an odd integer, say 11. Then adding 1 to that value of n gives you an
answer of 12 ... and that is an even answer. Therefore, n + 1 does necessarily return an odd
answer for every value of n.
.
Answer h is 3n. Well, suppose n is 8. Then 3 times n is 24, and that is even. Therefore,
3n is not necessarily an odd value for every value of n.
.
Answer i is 3n + 1. If n is 5 then 3n is 15. When you add 1 to that you get 16, an even answer.
Therefore, 3n + 1 is not necessarily an odd value for every value of n.
.
Hope that this helps you to understand the problem a little better.

You can put this solution on YOUR website!
Given:
.

.
When f(x) = 87 this becomes:
.

.
Transpose this equation (just switch sides) so the unknown x is on the left side. This transposition
makes the equation become:
.

.
Get rid of the 12 on the left side by subtracting 12 from both sides to get:
.

.
Next, get rid of the multiplier 3 on the left side by dividing both sides by 3 to reduce
the equation to:
.

.
On the left side of the equation, you can simplify it by applying the power rule of exponents
to both the numerator and the denominator of the fraction to get:
.

.
But for the numerator you can apply the rule that 1 raised to any power is just 1. When you
apply that rule, the equation becomes:
.

.
Get rid of the denominator by multiplying both sides of this equation by
and you have:
.

.
But 125 can be replaced by and 25 can be replaced by . When you do that you get:
.

.
Apply the power rule by raising the to the power to get:
.

.
Multiply out the exponent and you get:
.

.
When you multiply two common bases that are raised to an exponent, the rule is that you
add the exponents. So in this case you add 3x - 6 and 2 to get the exponent of 5 and the equation
then becomes:
.

.
Simplify the exponent by combining the -6 and the +2 and you have:
.

.
Now here's a little tricky insight. If you raise any number to the zero power the answer
is 1. So if you raise 5 to the zero power it will equal the 1 on the left side of this equation.
.
Therefore, we can say that the exponent 3x - 4 must equal zero. In equation form, this is:
.

.
Get rid of the -4 on the left side by adding 4 to both sides of this equation to get:
.

.
And solve for x by dividing both sides of this equation by 3 and you have the answer:
.

.
A little bit of work to get to this point. Hope this helps you to understand how to do the problem.
.

You can put this solution on YOUR website!
Check your answer again.
.
You are given the function:
.
y = 4x - 7
.
To find the inverse of this, replace y with x and x with y to get:
.
x = 4y - 7
.
Solve this for y. Begin by transposing the equation (just swapping sides to get the term
containing y on the left side). After transposing, the equation is:
.
4y - 7 = x
.
Get rid of the -7 on the left side by adding 7 to both sides to get:
.
4y = x + 7
.
Then solve for y by dividing both sides of this equation by 4 to get:
.
y = (1/4)x + 7/4
.
This is the inverse of the function y = 4x - 7.
.
That may have been the answer you got. If so you should have written it as:
.
y = (x + 7)/4
.
so that it was clear that both terms, the x and the 7, were to be divided by 4. According to
the rules of algebra, the way you wrote the answer, only the 7 was divided by 4 and x was then
added to the result of that division so that the answer was x + (7/4).
.
Now, how do you tell whether or not that the inverse is a function? Look at the answer:
.
y = (1/4)x + (7/4)
.
This is in the form of a slope intercept equation ... the slope is (1/4) and the y-intercept
is at +7/4 on the y-axis. The graph is a straight line. ... And the graph looks like this:
.

.
One way to tell if this is a function is if you can draw a vertical line anywhere on
the graph and it only intersects the graphed line at a maximum of one point, the graphed line
represents a function. On the graph shown above, anywhere you draw a vertical line it will cross the
graphed line only at one point, so you have a function. If somewhere on the graph a
vertical line could cross the graphed line at more than one point, then the graph would
not represent a function.
.
So the answer to your problem is yes ... the inverse is a function.
.
Hope this helps you to understand one way of determining whether an expression represents
a function.
.

You can put this solution on YOUR website!
The problem tells you that the price of premium is 5% more (or 0.05 times) than the price
of regular.
.
Multiply the price of regular ($1.60) by 0.05 and you get $0.08 cents more. Then just add
the $0.08 to the price of regular and you get that the cost of premium is $1.68 per gallon.
.
Another way you can look at this problem is to say that the price of premium is 105% more
than the price of regular. 105% is the same as 1.05. So multiply the price of regular times
1.05 and you get that $1.60 times 1.05 = $1.68 per gallon. Same answer, just a little
different way of viewing the problem.
.
Just curious ... how old is your math text? And where in the US can you buy gasoline
nowadays for $1.60 per gallon?
.
Hope this helps you to understand the problem a little better.
.

You can put this solution on YOUR website!
If X is the unknown number, then the sum of that number and 8 is represented as X + 8. And twice
the sum of the number and 8 is 2(x + 8). The problem tells you that this must equal 40.
In equation form this is:
.
2(X + 8) = 40
.
Multiply out the left side by multiplying 2 times each of the terms in parentheses to get:
.
2X + 16 = 40
.
Get rid of the 16 on the left side by subtracting 16 from both sides to get:
.
2X = 24
.
Solve for X by dividing both sides of this equation by 2 and you have that X, the unknown
number, is:
.
X = 24/2 = 12
.
Check: Does twice the sum of 12 and 8 = 40??? 12 + 8 equals 20 and twice 20 is, in fact,
40. So the answer checks. X = 12 is the answer.
.
Hope this helps you to understand the problem.
.

You can put this solution on YOUR website!
Consecutive even numbers can be represented by x for the first and the next number has to be
2 more than that. So the next even number is x + 2. (Think of 6 and 8 as being consecutive
even numbers ... and note that they are 2 units apart.)
.
You are now told that the product of these two consecutive even numbers is 48. So multiply the
two numbers together and set the result equal to 48 as follows:
.

.
Multiply out the left side and this equation becomes:
.

.
Get this into standard quadratic form by subtracting 48 from both sides and you have:
.

.
The left side of this equation can be factored as follows:
.

.
Notice this equation will be true if either of the two factors on the left side equals zero
because a multiplication by zero will make the entire left side become zero and therefore
equal to the right side.
.
Set the two factors equal to zero (one at a time) as follows:
.

.
Solve for x by subtracting 8 from both sides to get:
.

.
then set the other factor equal to zero to get:
.

.
Solve by adding 6 to both sides:
.

.
So there are two possible answers to this problem because x (the first of the consecutive
even integers) can either be -8 or + 6.
This means that in the first possible set of consecutive even integers you have -8 and as the
first integer and therefore the next consecutive integer is 2 greater than that or -6. These
numbers are consecutive and even and their product is +48.
.
For the second answer you have that the first integer (x) is +6 and therefore the next
consecutive integer is +8. This pair (+6 and +8) are consecutive and even and their product
is also +48.
.
So the two answers are either -8 and -6 or +6 and +8.
.
Hope this helps you to see how this problem can be worked.
.

You can put this solution on YOUR website!
Let D represent the number of ducks and T represent the number of turtles.
.
The problem tells you that there are 13 animals. This means that if you add the number of
ducks to the number of turtles the sum is 13. In equation form this is:
.
D + T = 13
.
Each duck has 2 legs, so to get the number of duck legs you multiply the number of ducks by 2
to get 2D. And each turtle has 4 legs, so to get the number of turtle legs you multiply the
number of turtles by 4 to get 4T. If you now add these two amounts together, the problem tells
you that the total is 40. In equation form this is:
.
2D + 4T = 40
.
You now have the two equations you need to solve for the number of ducks and turtles.
.
Return to the first equation and solve it for one variable in terms of the other. For
example, let's solve for D in terms of T by subtracting T from both sides of this equation
to get:
.
D = 13 - T
.
Then go to the second equation and substitute 13 - T in place of D. This substitution
makes the second equation become:
.
2(13 - T) + 4T = 40
.
Do the distributed multiplication on the left side by multiplying 2 times each of the
terms inside the parentheses to get:
.
26 - 2T + 4T = 40
.
Get rid of the 26 on the left side by subtracting 26 from both sides and you have:
.
-2T + 4T = 14
.
Combine the two terms on the left side and this reduces the equation to:
.
2T = 14
.
Solve for T by dividing both sides of this equation by 2:
.
T = 7
.
You now know that there are 7 turtles and since there are 13 animals, the remaining
6 animals must be the ducks.
.
Hope this helps you to see where your answer of 6 ducks and 7 turtles came from.
.

You can put this solution on YOUR website!
Given the following pair of equations:
.
8x - 4y = 16
y = 2x - 4
.
To solve a pair of equations, you need to solve one of the equations for one variable in
terms of the other variable. Then you substitute that equivalent value into the other equation
and solve it.
.
Notice that the second equation has already been solved for y in terms of x. Therefore,
you already know that y is equal to 2x - 4.
.
Now go to the first equation and substitute that value (2x - 4) for y in that equation.
When you do that substitution the first equation becomes:
.
8x - 4(2x - 4) = 16
.
You now have an equation that contains only one variable, so it can be solved to get that
variable. Begin the solution by multiplying the -4 times each of the terms in the parentheses.
When you do that multiplication the left side of the equation becomes:
.
8x - 8x + 16 = 16
.
Notice that the 8x and the -8x cancel each other out and the equation reduces to:
.
16 = 16
.
Well, that is an interesting development. What does it mean. To find out, let's go back
to the original pair of equations. Let's take the first equation:
.
8x - 4y = 16
.
and let's try to work it into the form of the second equation ... with just a y on the left
side and everything else on the right side. Begin by subtracting 8x from both sides to
get rid of the 8x on the left side. When you do this subtraction the equation becomes:
.
-4y = -8x + 16
.
Now divide both sides of the equation (all terms) by -4 so that you just have y on the left
side. When you divide all terms on both sides by -4 the equation becomes:
.
y = 2x - 4
.
Wow! Notice that this is the exact same equation as the second equation. Therefore,
if you graphed both equations you would find that the two graphs coincide ... one graph is
identical to the other graph. This means that there is not a unique solution to the two original
equations. Every solution pair of one equation is also a solution pair of the other equation.
.
Let's try a point just to verify this. For example, suppose in the first equation we
assume that x = 0. This means that in the equation
.
8x - 4y = 16
.
we set x equal to zero and the equation reduces to
.
-4y = 16
.
Divide both sides by -4 and the equation reduces to
.
y = -4
.
This tells us that the point (0, -4) is on that graph
.
Now go to the second equation and set x = 0. The second equation equation is
.
y = 2x - 4
.
and when you set x = 0 it reduces to
.
y = -4
.
This tells you that the coordinate point (0, -4) also in the solution set for this equation.
.
You can do the same sort of exercise for other values of x and you will always find the
same thing to be true. For example if you set x equal to 5 in both equations you will find the
corresponding value of y will be 6. Therefore, the point (5, 6) is a solution for both of
the equations. There are an infinite number of common solutions, not a just a single solution.
.
In working this problem you came up with the same solution as I did. And you were correct.
You just needed to find out what this meant when you worked the two equations out and found
they were identical. Rest easy ... you know what you are doing.
.
Hope this helps you to understand what was going on in this problem.

You can put this solution on YOUR website!
Consecutive odd integers are 2 digits apart (think of 3 and 5 and 7 as being consecutive
odd integers). So if we let x represent the first odd integer, the next consecutive odd
integer is x + 2 and the next consecutive odd integer after x + 2 is x + 4.
.
Then three times the third integer is 3*(x + 4) which multiplies out to 3x + 12.
.
The problem tells you that the sum of the three consecutive odd integers is 3 times the
third integer. In equation form this can be written as:
.
x + (x + 2) + (x + 4) = 3x + 12
.
On the left side of this equation combine the like terms to get:
.
3x + 6 = 3x + 12
.
What happens if you subtract 3x from both sides? This reduces the "equation" to
.
6 = 12
.
And that obviously cannot be and is not true. This tells you that there is no value of
x that will satisfy this problem. That further means that there are no 3 consecutive
odd integers that add together to give you a sum equal to 3 times the third of the three
consecutive odd integers.
.
Hope this helps you to understand why you were having a problem with this exercise.
.

You can put this solution on YOUR website!
At the end of the first day Adam will owe Gerry the $20 plus 2% (or .02) of $20. So at the
end of the first day Adam owed Gerry:
.
20*(1 + .02) = 20*(1.02)
.
At the end of the second day Adam owed what he owed on the previous day [20*(1.02)] times
(1.02) again or Adam owed Gerry:
.
(20*(1.02))*(1.02) = 20*(1.02)^2
.
At the end of the third day Adam owed what he owed on the second day times 1.02 again. So
Adam owed:
.
(20*(1.02)^2)*(1.02) = 20*(1.02)^3
.
By now you might see the pattern. The amount owed on each day is the original 20 times 1.02
raised to the power equal to the number of days the money has been on loan. For example,
at the end of January, the money has been on loan for the 31 days in January so the amount
owed on January 31st is:
.
20*(1.02)^31
.
and a little calculator work will tell you that this amount is $36.95 rounded to the nearest
cent.
.
On February 18th the money will have been out on loan for 49 days (31 days in January and
18 days in February is a total of 49 days). This means that the amount of money Adam will
own is:
.
20*(1.02)^49. A calculator will tell you that 1.02^49 = 2.638811793 and when you multiply that
by $20 you get an answer of $52.78 rounded to the nearest cent.
.
Do you realize that if this continued for 365 days (one year) Adam would have to pay Gerry
back $27,548.17 ... this comes from 20*(1.02)^365.
.
In real life this is called "loan-sharking". At first it doesn't sound too bad. If you borrow
$20 for a single day, you would have to pay back $20.40. But when you get figuring it out
you can see that the longer you go without paying it back just how expensive it can get to be.
What a rip-off this is ... (bad news for Adam but very profitable for Gerry.)
.
In summary, the answer to this problem is $52.78. Hope this helps you to see how the problem
can be worked.
.

You can put this solution on YOUR website!
Well, you can be sure ... you are correct.
.
One way to look at it is to start with the given equation:
.
y = (1/2)^x
.
Then let's just start with x = 0 and increment it by 3 each step.
.
When x = 0 then y = (1/2)^0 = 1 ... (any number to the zeroth power equals 1)
.
When x = 3 then y = (1/2)^3 = 1/8
.
When x = 6 then y = (1/2)^6 = 1/64
.
When x = 9 then y = (1/2)^9 = 1/512
.
Each time x increments by 3 you need to multiply the previous term by 1/8 to get the next
term. So you are correct. Good job and keep up the good work.
.

You can put this solution on YOUR website!
A quick way to graph this is to first set y equal to zero and solve for x. When you get that
answer for x, it will be on the x-axis because any point on the x-axis has zero for its
corresponding value of y.
.
Then you next set x equal to zero and solve for y. The value you get for y will be on the y-axis
because any point on the y-axis has zero as its corresponding x value.
.
You then have two points on the graph ... a point on the x-axis and a point on the y-axis. You
can then extend a straight line through these two points and you have the graph.
.
Let's do it. Start with the given equation 2y = x + 4
.
Set y = 0 and this equation becomes:
.
0 = x + 4
.
Solve for x by subtracting 4 from both sides to get:
.
-4 = x
.
Mark the point -4 on the x-axis.
.
Then set x = 0 and the equation you were given and the equation becomes:
.
2y = 0 + 4
.
Solve for y by dividing both sides by 2 to get:
.
y = +2
.
Mark the point +2 on the y-axis
.
Draw a straight line through these two points and you should have a graph that looks like this:
.

.
You should check at least one other point on the graph just to make sure that you didn't
make a mistake in calculating one of the points. For example, let's just choose x = +3
.
Substitute +4 for x in the given equation and it becomes:
.
2y = 4 + 4
.
Solve for y and you get
.
2y = 8
.
y = 8/2 = 4
.
So you have determined that when x = 4, then y = 4. Go to 4 on the x-axis, and go up 4 units
in the y direction to see if that point (4,4) is on the graph. It appears to be, so the graph
is likely to be correct. You can try a couple of other values for x if you would like and
see it the (x, y) points you get are also on the graph ... just to verify that your first
two points on the x and y-axes were correct.
.
Hope this helps you to understand the problem and how to work it.
.

You can put this solution on YOUR website!
An analytical way of doing this problem is to look at the factors of 120.
.
Divide it by 2 and you get 60*2
.
Divide the 60 by 4 to get 4 and 15 as factors of 60 and you have that the factors of 120
have become 4*15*2.
.
Then factor 15 into 5*3 and substitute that result for 15 to get that the factors of 120
are 4*3*5*2. If you multiply these factors together you do get 120, so our factors check out
to be OK.
.
Now arrange the factors in ascending order: 2*3*4*5
.
Notice significantly that each of these factors increases by 1. Now compare them to the
factors on the left side of the problem you were given ...
.
(x+1)*(x+2)*(x+3)*(x+4)
.
Each of those factors is 1 greater than the preceding factor ... and there are four of them
just as we found in the four factors 2*3*4*5
.
What do we get if we set the first factor x + 1 equal to 2 ... we find x = 1
.
What do we get if we set the second factor x + 2 equal to 3 ... we again find x = 1
.
What do we get if we set the third factor x + 3 equal to 4 ... we again find x = 1
.
And, finally, what do we get if we set the fourth factor x + 4 equal to 5 ... we again find x = 1
.
What an interesting outcome. If in each of the four factors:
.
(x+1)*(x+2)*(x+3)*(x+4)
.
we set x = 1, the factors become 2*3*4*5 and this product is 120. So the solution
to this problem is x = 1
.
Sure was a lot easier to do it this way than to try multiplying out the 4 original factors
and then setting that product equal to 120 and trying to solve that fourth order equation.
.
Hope this is clear enough so that you can follow it through and see what was done to solve
the problem.
.

You can put this solution on YOUR website!
To solve this problem, you need to begin by finding two equations.
.
First, let's let F represent the first number and S represent the second number.
.
The first sentence of the problem tells you that if you add these two numbers, the sum is 89.
So adding them results in the equation:
.
F + S = 89
.
That's one equation. The you are told that adding 4 times the first (or 4F) to the second (or S)
gives 155. So you get the second equation by adding these quantities as follows:
.
4F + S = 155
.
You can't solve a linear equation by itself if it contains two unknowns. Therefore, you
need to figure out a way of eliminating one of the unknowns in one of the equations.
.
Suppose you go to the first equation and you subtract F from both sides. This results in:
.
F + S - F = 89 - F
.
On the left side the F and the -F cancel each other and you are left with the equation:
.
S = 89 - F
.
Next, go to the second equation:
.
4F + S = 155
.
You now know that S = 89 - F so you can substitute 89 - F for S in the second equation to
get:
.
4F + 89 - F = 155
.
This new equation only has one variable ... F ... so you can solve it.
.
On the left side, combine the 4F and the -F to get 3F. This reduces the equation to:
.
3F + 89 = 155
.
Then get rid of the 89 on the left side by subtracting 89 from both sides and you have:
.
3F = 66
.
Solve for F by dividing both sides by 3 and the result is:
.
F = 66/3 = 22
.
So now you know that F = 22. You can then return to either of the original equations,
substitute 22 for F, and solve for S. Let's go back to the first equation:
.
F + S = 89
.
Substitute 22 for F and this equation becomes:
.
22 + S = 89
.
Solve for S by subtracting 22 from both sides:
.
S = 89 - 22 = 67
.
So the answer is that the 2 numbers are 22 and 67.
.
You can further check by substituting these numbers into the original second equation.
When you do you get:
.
(4*22) + 67 = 88 + 67 = 155
.
and this is what the second part of the problem tells you it should be. Therefore,
the first number of 22 and the second number of 67 check out. This answer is correct.
.
Hope this helps you to work your way through the problem and understand it a little better.
.

You can put this solution on YOUR website!
A little "tricky". The answer to this question is that any real value of x (positive or
negative) will satisfy this equation. Why? Because x^2 will always be a positive number. And
when you add 9 to it, the result is still a positive number. And a positive number is greater
than zero.
.
Let's just try a few values of x to demonstrate this.
.
Suppose x is +10. Then x^2 + 9 = 10^2 + 9 = 100 + 9 = +109. This certainly is greater than
zero as is to be the case required by the problem.
.
Suppose x is +1. Then x^2 + 9 = 1^2 + 9 = 1 + 9 = +10. This is still greater than zero.
.
Suppose x equals zero. Then x^2 + 9 = 0^2 + 9 = 0 + 9 = +9. Still greater than zero.
.
Now suppose x = -1. Then x^2 + 9 = (-1)^2 + 9 = 1 + 9 = +10. Still greater than zero.
.
Then suppose x = -10. Then x^2 + 9 = (-10)^2 + 9 = 100 + 9 = +109. And it is also greater
than zero.
.
In fact if we graph y = x^2 + 9 the graph looks like:
.

.
You can see from this graph that x^2 + 9 never has a value less than 9 no matter what the
value along the x axis is. Therefore, x^2 + 9 is always greater than zero.
.
Hope this helps you to understand the problem a little better.
.

You can put this solution on YOUR website!
Given:
.

.
To simplify this begin by factoring 112. First step, since 112 is even divide by 2 and you have
112 = 56*2. But the 56 is even, so you can also divide it by 2 and the 56*2 then becomes
28*2*2. And since the 28 is also even, divide it by 2 and the 28*2*2 becomes 14*2*2*2.
And again, the 14 is even so divide it by 2 and the final form is 7*2*2*2*2. The 7 is a prime
number so you cannot factor it down further.
.
Return to your original expression and substitute 7*2*2*2*2 for 112 and you have:
.

.
You can split this up into the products:
.

.
But
.
Substitute 2 for each of the terms and you reduce the expression to:
.

.
Multiply the three 2's together to get 2*2*2 = 8 and the problem reduces to:
.

.
So congratulations ... your answer was correct. Good job! Keep up the good work.
.

You can put this solution on YOUR website!
You are given:
.
h(t) = 5 - t^2
.
and you are to find h(2/3). All this is telling you to do is to go to the given function
for h(t) and for both sides wherever you see a "t" you substitute 2/3. Then you just
simplify the terms.
.
In this problem you start with the function:
.
h(t) = 5 - t^2
.
for every t in this function you substitute 2/3 and you get:
.
h(2/3) = 5 - (2/3)^2
.
Next, on the right side square the 2/3 and you get 4/9. Substitute 4/9 for (2/3)^2 and
the result is:
.
h(2/3) = 5 - 4/9
.
Finally, subtract 4/9 from 5 [equivalent to subtracting 4/9 from 4 9/9] and you get the
answer 4 5/9 ... that's read as "4 and 5-ninths" and it's equivalent to 41/9.
.
So the answer to this problem is:
.
h(2/3) = 4 5/9 = 41/9
.
Hope this helps you to understand the process of evaluating a function and what the notation
means. In this case h(2/3) means "find the value of h(t) when t equals 2/3."
.

You can put this solution on YOUR website!
The first thing to do is to check the x^2 term and make sure that its coefficient (multiplier)
is 1. In this problem it is, so you can proceed.
.
Next look at the coefficient (multiplier) of the x term. That coefficient is -4. Then the
rule is to divide that by 2, square that, and add that result to the binomial you were originally
given. The result will be a perfect-square trinomial. Let's do it.
.
Start with the multiplier of the x term. That multiplier is -4. Divide it by 2 and you get
an answer of -2. Square the -2 and you get +4. Add that to the original binomial you were
given and it becomes:
.
x^2 - 4x + 4
.
And this is the perfect-square trinomial you were to find.
.
Notice that this can be factored as follows:
.
x^2 - 4x + 4 = (x - 2)(x - 2) = (x-2)^2
.
How can you easily tell what the factors will be? They will be x followed by half of the
original multiplier of the x term in the binomial ... in this case x followed by half of
the -4 that multiplied the x term. So the factors are both (x - 2).
.
Hope this gives you some insight into the process of making a perfect-square trinomial
when you are given the first two terms that involve x^2 and x.
.

You can put this solution on YOUR website!
The given expression falls under the following factoring rule:
.

.
In the given expression note that X = 2a because when you square both sides you get
.
Also note that Y = 6b because when you square both sides you get
.
Substituting 2a for X and 6b for Y in the factoring rule leads to:
.

.
Then notice on the right side that each of the two factors has 2 as a common factor of the
two terms in parentheses. This means that each factor on the right side can have a 2 factored
from it, and this leads to:
.

.
Multiply the two factors of 2 to get 4 and you have the final answer of:
.

.
If you like, you can then square out the two terms on the left side and you have:
.

.
Hope this helps you understand how to do the problem, and how to use the rule that applies.
.

You can put this solution on YOUR website!
The words "the number" tells you that there is some unknown number. Let's call that unknown
number "x". Next you are told that this unknown number is to be increased by 18. That means
you are to add 18 to x. In algebraic form this is written as:
.
x + 18
.
Finally, you are told that this algebraic form is to equal 60. In equation form this becomes:
.
x + 18 = 60
.
You can interpret this as some number (x) plus 18 results in 60. From algebraic procedures
we can solve for x by getting rid of the 18 on the left side of the equation. We do this
by subtracting 18 from both sides of the equation ... to get:
.
x + 18 - 18 = 60 - 18
.
Notice on the left side that the +18 and the -18 cancel each other out, so on the left
side only x remains. On the right side the 60 take away 18 results in 42. So the equation
you are left with is just:
.
x = 42
.
This is the answer to the problem. What number if you add 18 to it gives you 60? The answer
is that the number is 42 because if you add 18 to 42 you get 60.
.
Hope this helps you to understand the problem.
.

You can put this solution on YOUR website!
What do the sides add up to???
.
You can't tell from the information given. Let me explain with a couple of examples.
.
Suppose the rectangle has sides of: length = 7 m and width = 6 m
.
If you multiply these two together you get an area of 42 m^2. And if you add the 4 sides you
get a total of 6 + 7 + 6 + 7 = 26 m
.
Next suppose the rectangle has sides of: length = 42 m and width = 1 m
.
If you multiply these two together you get an area of 42 m^2. But if you add the 4 sides you
get a total of 1 + 42 + 1 + 42 = 86 m
.
This shows that you can't be sure of what the sides are if you are just given the area.
.
Hope this helps you to see that you can't tell the total of the perimeter (total of the sides)
if you are just given the area of a rectangle. You need some other information.