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This is a "trick" problem to see how observant you are ... You are given the two equations:
.
2x+y=1
y=1-2x
.
Let's rearrange the bottom equation by adding 2x to both sides. When you do that the bottom
equation becomes 2x+y = 1. But that is identical to the top equation. Therefore, the bottom
equation and the top equation are identical. Every solution for the top equation is a solution
for the bottom equation also. They are not independent equations and therefore they do
not have a single, unique solution.
.
Hope this helps you to see your way through the problem and why the answer is what it is ...
.

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A prime number is a number that can only be divided by itself or 1 to get an integer as
an answer. For example is 3 a prime number? Yes it is because it can only be divided
by 3 and 1 to get an integer as an answer. How about 6 ... is it a prime number? No it is
not because it can be divided by 6, by 1, by 3, and by 2 and each of those divisions
results in an integer as an answer.
.
You are to find all the prime numbers between 55 and 75. Note that both 55 and 75 are not
prime numbers because both can be divided by 5.
.
56 is not prime ... it can be divided by 2. From this you can deduce that any even number (except 2)
is not a prime because it can be divided by 2. Therefore, only odd numbers can be a prime
and every odd number is not a prime. And any number that ends in 5 is not prime because it
can be divided by 5 and produce a quotient that is an integer. (2 is an exception
to this rule. It is prime because even though it is even, it can only be divided by itself
and 1 ... so it fits the definition of prime.)
.
The prime numbers between 55 and 75 are: 59, 61, 67, 71, and 73. Any other numbers in
this range are divisible by something other than themselves and 1. Any of the other numbers
in this range have at least a 2 or a 3 or a 5 as a divisor.
.
Hope this clarifies the term "prime" for you and helps you to see your way through the
problem.
.
.

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Given that the value of a car (V) that is t years old is given by the equation:
.
V = -2000t + 21,000
.
The problem tells you that the value of a car is $11,000 and asks you how many years old
the car is ... where t is the age of the car. Substitute 11000 for V in the equation to
get:
.
11000 = -2000t + 21000
.
Get rid of the 21000 on the right side by subtracting 21000 from both sides to change the
equation to:
.
-10000 = -2000t
.
Now solve for t by dividing both sides of this equation by -2000 and you have:
.
-10000/-2000 = t
.
Doing the division on the left side results in:
.
5 = t
.
The car is 5 years old.
.
Hope this helps you to understand this problem and how to find the answer.
.

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Think about this ...
.
To cut a log into 4 pieces you need to make 3 cuts as follows:
.
xxxxx-cut-xxxxx-cut-xxxxx-cut-xxxxx
.
The strings of 5 x's represent the pieces. And the labels "-cut-" show where the cuts
are made. Therefore, with three cuts you create 4 pieces from the log. Since the total
time of the cuts is 12 seconds, the 3 cuts take 4 seconds each.
.
To cut the log into 5 pieces you need 4 cuts as follows:
.
xxxxx-cut-xxxxx-cut-xxxxx-cut-xxxxx-cut-xxxxx
.
Since each cut takes 4 seconds, the time for the 4 cuts is 4*4 = 16 seconds.
.
Hope this helps you to visualize the problem and see how to get the answer.
.

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Given:
.
+6x+3y=27
-4x+7y=27
This is what I started, but I'm not sure if it's right....
You have the right idea for solving this equation by eliminating one of the variables.
To use this method you need to get one of the variables in one equation to equal the same
variable term in the other equation. You can do this by multiplying both equations by an
appropriate number. You have chosen to eliminate the variable y terms. And you are multiplying
the top equation by +7 and the bottom equation by -3. This work is shown below:
+7(+6x+3y=27) = 42x+21y = 189 <== OK
-3(-4x+7y=27) = 12x-21y = -81 <== note you had +81 and I changed it to -81 because -3*27 = -81
.
You now have two equations:
.
42x + 21y = 189 and
12x - 21y = -81
.
Now you can add these two equations vertically. (You are adding equals to equals, so
the resulting equation is still balanced.) When you add vertically, 42x + 12x equals 54x.
And next, +21y added to -21y results in a cancellation ... so the y terms disappear.
Finally, on the other side, 189 added to -81 is 108. So the vertically adding of the two
equations results in:
.
54x = 108
.
Solve for x by dividing both sides by 54 to get:
.
x = 108/54 = 2
.
You now know that x is 2. You can return to either of the two equations that you were given
originally and in the one you select you can substitute 2 for x and then solve for y.
.
Let's return to the original equation 6x + 3y = 27. If we substitute 2 for x the term 6x
becomes 6*2 = 12. So the equation becomes:
.
12 + 3y = 27
.
Get rid of the 12 on the left side by subtracting 12 from both sides to reduce the equation
to:
.
3y = 15
.
Finally solve for y by dividing both sides of this equation by 3 to get:
.
y = 15/3 = 5
.
In summary, the answer to this problem is x = 2 and y = 5.
.
What does this tell you? It tells you that the (x, y) point [which is (2, 5)] satisfies
both equations. That is, if you go to the two equations you were given and let x equal 2 and
y equal 5, both sides of each of the two equations are equal. It also tells you that if you
graph both equations, the two graphs will cross at the point (2, 5).
.
Hope this helps you with understanding the problem and how to get the answer. You started
out on the right track.
.

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Given:
.

.
You are asked to find the domain of this function. The domain is all the values that x
can take.
.
In this problem x can take any real value except x cannot equal -4. Why can't it be -4?
.
The answer is that if x is -4, then the denominator of the given fraction becomes zero ...
-4 + 4 = 0. And since the numerator (which is 1) is divided by the denominator, this means
that 1 would be divided by zero if x is equal to -4. But the rules of algebra tell you
that division by zero is not allowed.
.
So the answer to this problem is that the domain for x is all real numbers from minus infinity
to plus infinity except x cannot equal -4.
.
Hope this helps you to understand the problem.
.

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The formula for this type of problem is:
.
Distance = Speed * Time
.
You might be able to understand this by thinking, "If I go at 60 miles per hour for 2 hours,
how far do I travel?" The answer is:
.
Distance = 60 miles per hour* 2 hours = 120 miles
.
In the problem you are told that the distance to be covered is 12 miles. The speed is 55 miles
per hour. Put these into the equation and you get:
.
12 miles = 55 miles per hour * Time
.
You can then solve for the Time by dividing both sides of this equation by 55 to get:
.
12/55 = Time
.
So it takes 12/55 hours to go 12 miles. The fraction 12/55 converts to a decimal by dividing
55 into 12 to get 0.218181818 of an hour. Since an hour has 60 minutes, you can convert
the answer to minutes by multiplying 60 by 0.218181818 to get about 13.09 minutes.
.
So at 55 miles per hour you go 12 miles in a little over 13 minutes.
.
Hope this helps you to understand the problem and how to use the formula to solve it.
.

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Let D equal 1 carton of dolls and T equal 1 carton of trucks. The problem says that the
combined numbers of cartons is 100. Therefore, if we add D and T the total number of cartons
is 100. In equation form this is:
.
D + T = 100
.
Since the dolls cost $140 per carton, if we multiply the number of cartons of dolls by
$140, we have the total amount of money spent on dolls. Therefore, $140N is the amount
spent on dolls.
.
Similarly, trucks are $430 per carton. Therefore, the total amount spent on trucks is $430
times the number of cartons of trucks that were purchased.
.
When you add these two amounts together the total is $28,500. In equation form this is:
.
140D + 430T = 28,500
.
But from the very first equation we set up (D + T = 100) we can subtract T from both sides
to get D = 100 - T. This means that we can substitute 100 - T for D in the "money"
equation. When we make this substitution, the money equation becomes:
.
140*(100 - T) + 430T = 28500
.
Doing the multiplication on the left side results in:
.
14000 - 140T + 430T = 28500
.
If you combine the two terms containing T on the left side you get 430T - 140T = 290T. This
makes the equation become:
.
14000 + 290T = 28500
.
Then get rid of the 14000 on the left side by subtracting 14000 from both sides to reduce
the equation to:
.
290T = 14,500
.
Solve for T by dividing both sides by 290 and you have:
.
T = 14500/290 = 50
.
This says that there are 50 cartons of trucks. Since the total number of cartons is 100, that
means that there also must be 50 cartons of dolls.
.
Check using the money relationship. 50 cartons of dolls at $140 per carton multiplies
out to be $7000. And 50 cartons of trucks at $430 per carton multiplies out to be $21500. Adding
these two amounts results in $7000 + $21500 = $28500 and that total is what the problem
says it should be. Therefore, the answer of 50 cartons of dolls and 50 cartons of trucks
is correct. Hope this helps you to understand the problem and how to get the answer.
.

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Given:
.

.
This means that y is the value of
.
The answer to this problem is that y is equal to zero, so you can say that
is equal to zero. Notice that you are saying that the log of 1 is equal to zero regardless
of what the base is.
.
Let's see why. We can do this by translating from the logarithmic form to its equivalent
exponential form. This translation says that:
.
has an equivalent exponential form
.
By comparing the log form of the translation to the log equation you were given in this
problem, you can see that:
.

and

.
Then you can see that by substituting the right side of these three relationships into
the exponential form, that the exponential form of the given problem is:
.

.
Now it's time to recall that if any base number is raised to the zero power, the answer is 1.
Since we need 1 as the answer, we need to raise g to the zero power. So y = 0 and this
means that equals zero.
.
Hope this helps you to understand this problem and how to get the answer.
.

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Given:
.

.
Note that if you solve for y, the answer is equal to
.
One way to solve this equation is to translate it to the exponential form of the logarithm.
The translation uses the relationship that:
.
is equivalent to the exponential form
.
By comparing the log form in the translation to the log form you were given in this problem
you can see that:
.

and

.
Substituting the values on the right side for b, A, and y in the exponential form you
get:
.

.
Looking at this equation, you can see that to make the left side equal to the right side
the exponents have to be equal. In other words, y has to be 9 for the two sides to be
equal. And since y is equal to , then you can say that equals 9.
.
Hope this helps you to understand the problem.
.

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You are given the equation:
.

.
To solve for x you can convert the given logarithmic equation to its equivalent exponential
form using the following translation format:
.
is equivalent to the exponential form
.
Note that by comparing the given equation to the translation format you can see from the
positions in the equations that:
.
,
and

.
Substituting values for b, A, and y into the exponential form results in:
.

.
You can then solve for x by taking the square root of both sides to get:
.

.
So the answer to your problem is
.
It's a good idea to become very familiar with switching back and forth between the logarithmic
and exponential forms. This switching is often used in solving logarithmic equations.
.
Hope this helps you to understand this problem.

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Given the points (-9,3) and (4,-2)
.
To find the slope M of the line joining these two points you can use the formula:
.

.
If you let the point {-9, 3) be the first point ... meaning its values of x and y are
and ... then the second point is (4, -2) which means that
its x and y values are and respectively. Then just plug these
values into their locations in the formula and you get:
.

.
So the slope is
.
You might be able to speed up the process and to do it in your head by just looking at
the points (-9, 3) and (4, -2) and say to yourself:
.
To go from the x value of the first point to the x value of the second point, I would go
from -9 to +4. This means that on the x-axis I would move from the left to the right 13
numbers or +13. At the same time, to go from the y value of the first point to the y-value
of the second point I would go from +3 to -2 in the y-direction. Since I am moving down
the change is negative, and the number of units to get from +3 to -2 is 5, so the change is
-5 units. This means the slope is -5 divided by +13 which again gives the answer
.
Hope this helps you to understand the problem a little better.

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See answer 76816 to problem 105578. Problem 105578 is the same as this problem (problem 105576).

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Let the age of Rex's father be F.
.
Then Rex's age (30 years younger than his father) is F - 30.
.
Rex's mother is 3 times as old as Rex. So her age is 3 times Rex's age or 3(F - 30).
.
Rex's father's age (which is F) is two years more than his mother's age (which is 3(F - 30).
So if you take 2 years from his father's age, the result will equal his mother's age. In
equation form this is:
.
F - 2 = 3(F - 30)
.
Multiply out the right side and you have:
.
F - 2 = 3F - 90
.
Get rid of the 2 on the left side by adding 2 to both sides:
.
F = 3F - 88
.
Eliminate the 3F on the left side by subtracting 3F from both sides to make the equation
become:
.
-2F = -88
.
Solve for F by dividing both sides by -2 and you have:
.
F = -88/-2 = 44
.
So Rex's father is 44 years old. Since Rex is 30 years younger than that, Rex is 44 - 30 or 14
years old. And since Rex's mothers age is 3 times Rex's age, his mother is 3 times 14 or
42 years old. Notice that Rex's father is two years older than Rex's mother, so the problem
checks.
.
In summary ... Rex's age is 14.
.
Hope this helps you to understand the problem.
.

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Given:
.

.
Inside the first set of parentheses, add the which is the same as
to get . Then square that quantity to get . This reduces the
problem to:
.

.
Multiplying the 8 times 2.25 results in 18 and the problem then is simplified to:
.

.
Next, multiply the 22 times each of the terms in the remaining set of parentheses. The
22 times the 1 is 22 and the 22 times the is . This makes the equation
become:
.

.
Then combine the 18 with the +22 and the +15 to get the sum 55. The equation is then:
.

.
By subtracting 55 from both sides you separate the two terms and make the equation:
.

.
Multiply both sides of this equation by t to eliminate the denominator and make the equation:
.

.
Solve for t by dividing both sides by -55 and you have:
.

.
Notice that both the numerator and the denominator have a common factor of 11. So you can
simplify the answer by dividing both by 11 to get:
.

.
Hope this helps you to understand the problem.

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Your answer is correct.
.
The method is:
.
Length = 5*w + 2
.
Width = w
.
Area = L * w = 65 sq cm
.
Multiply L * w and set it equal to the area to get the equation:
.
(L * w) = 65
.
Substitute 5*w + 2 for L to change the equation to:
.
(5w + 2)*w = 65
.
Multiply out the left side:
.

.
Subtract 65 from both sides:
.

.
Apply the quadratic formula to solve for w. When you do you get:
.

.
The term in the radical simplifies to:
.

.
Substituting this for the radical results in:
.

.
If you let the +- sign be negative you get a negative answer for w. That doesn't make sense
so we just use the + sign to get:
.
and this rounds to 3.41 cm.
.
This agrees with your answer, so you are correct.
.

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Given:
.

.
The numerator has two factors ... and
.
The denominator also has two factors ... and
.
The way to do this problem is to divide the factor in the denominator into the
factor in the numerator. The result of this division is .
.
Then divide the factor in the denominator into the factor in the
numerator. Since these two factors both have the base the division just involves
subtracting the exponent of the denominator from the exponent of the numerator. The result is:
.
.
.
Combining these two results leads to the answer of and this is in scientific
notation.
.
Hope this helps you to see how to work the problem.
.

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N divided by 7 is:
.

.
This is equal to negative 2 ... which is -2.
.
Putting this all together results in the equation:
.

.
Hope this helps you to understand the problem.

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You can solve this problem using proportions that are set up as follows:
.

.
The height of the woman is 5 ft 6 inches and this is 5.5 feet. Her shadow is 12 ft long.
Substituting these values results in:
.

.
The shadow of the tree is 45 feet. Substitute this and you get:
.

.
You can get rid of the denominator of 45 by multiplying both sides of the equation by 45.
When you do that the equation becomes:
.

.
You are now left with the equation:
.

.
Solve for the Height of the tree by multiplying out the numerator of 45*5.5 and dividing that
product by 12 to get:
.

.
The tree is 20.625 feet tall. This converts to 20 ft 7.5 inches.
.
Hope this helps you to understand the problem a little better ...
.

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Let L represent the length of the rectangle and W represent the width of the rectangle.
.
The area of the rectangle is to be 7500 square meters.
.
The area of a rectangle equals the product of the length L and the width W or:
.
A = L * W
.
Now substitute 7500 for the area to change the equation to:
.
7500 = L * W
.
The 250 meters of fencing is a little tricky. This fencing will only be needed on 3 sides of
the rectangle because the 4th side is an existing fence. So we can say that the fencing material
will be used for the Length and the two Widths to be installed, and the existing fence will
be used for the other Length. So the 250 meters of fence will equal L plus 2W. In equation
form this is:
.
250 = L + 2W
.
Now return to the area equation. We can solve it for W by dividing both sides by L to get:
.
W = 7500/L
.
Substitute the right side of this for W in the fence length equation to get:
.
250 = L + 2W = L + 2(7500/L) = L + 15000/L
.
So start with
.
250 = L + 15000/L
.
You can get rid of the denominator by multiplying both sides of this equation (all terms)
by L to make the equation become:
.
250L = L^2 + 15000
.
To get this in "standard form" for solving, make the left side equal zero by subtracting
250L from both sides to change the equation to:
.
0 = L^2 - 250L + 15000
.
A little more standard form is obtained by simply transposing sides to get:
.
L^2 - 250L + 15000 = 0
.
The left side of this equation can be factored to:
.
(L - 150)(L - 100) = 0
.
This equation will be true if either of the factors equals zero ... because a multiplication
by zero on the left side will make the left side of the equation equal the zero on the
right side.
.
Setting each factor equal to zero means:
.
L - 150 = 0 or L = 150
.
and L - 100 = 0 or L = 100
.
Let's first choose L = 150 meters.
.
That leaves 100 meters of fencing material (250 meters - 150 meters) to use in making the
2 widths. So each of the widtha is 100 meters divided by 2 or 50 meters.
.
So this solution is to make the rectangle 150 meters by 50 meters which multiplies out
to give an area of 7500 square meters and use the 250 meters of fence as 2 widths of
50 meters each and a length of 150 meters. 150 meters of the existing fence is used for
the 4th side
.
How about our other potential solution of L = 100 meters? That means that if you cut 100 meters
of fence for the length, you have 150 meters left over for the 2 widths. So each width
is 75 meters. This makes the dimensions of the garden 100 by 75 meters. This means that
the area is 100 times 75 which is 7500 square meters. And the 100 meters plus 75 meters plus
75 meters for the sides adds up to the 250 meters of fencing. The existing fencing can
supply the missing 100 meter length.
.
So there are two answers to this problem. One is to make the rectangle 50 by 150 meters,
using the existing fence to provide one of the 150 meter sides. The other is to make
the rectangle 75 by 100 meters, using the existing fence to provide one of the 100 meter
sides.
.
Hope this helps you to understand the problem.
.

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Let L represent the length and W represent the width.
.
You are told that L equals 2 inches more than 2W. In equation form this is:
.
L = 2W + 2
.
Next you are told that the perimeter (P) is 34 inches. The perimeter is found by adding all
the sides. In other words:
.
P = L + W + L + W
.
By combining like terms on the right side this can be simplified to:
.
P = 2L + 2W
.
Substitute 34 for the perimeter to get:
.
34 = 2L + 2W
.
But since L = 2W + 2 we can substitute 2W + 2 for L in the perimeter equation to get:
.
34 = 2(2W + 2) + 2W
.
Multiply out the right side by finding the product of 2 and each of the terms inside
the parentheses. When you do, the equation becomes:
.
34 = 4W + 4 + 2W
.
Get rid of the 4 on the right side by subtracting 4 from both sides to get:
.
30 = 4W + 2W
.
Combine terms on the right side to reduce the equation to:
.
30 = 6W
.
Solve for W by dividing both sides by 6 to get:
.
5 = W
.
So we know the width is 5 inches. But our first equation tells us that the Length is
twice the width plus 2 inches ... or 2 times 5 and then add 2. This makes the length 12
inches.
.
In summary, the width is 5 inches and the length is 12 inches.
.
Check by adding the 4 sides: 5 + 12 + 5 + 12 to get 34 inches for the perimeter, just as
it should be.
.
Hope this helps you to understand the problem and how to solve.
.

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Let's define Point 1 as (-7,5) and Point 2 as (2, -3).
.
Then you can say that and
.
Similarly you can say that and
.
Then the equation for the slope (M) is:
.

.
Now all you have to do is substitute the values identified above to get:
.

.
So the slope is
.
Hope this helps you to see your way through the problem.
.

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If you factor the left side you get:
.

.
Then if you take the square root of both sides you get:
.

.
Note the right side has a plus and minus sign. If you now subtract 2 from both sides
to get rid of the 2 on the left side you get:
.

.
Hope this helps you to find the source of your difficulty.
.

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You start with:
.
f(x) = x^2
.
You can rotate it about the x-axis by changing the sign of the function. Call the new function
g(x). So for the first translation (about the x-axis), we have:
.
g(x) = -x^2
.
Next you want to shift the graph down by 5 units. Do this by just subtracting 5 from the
function. So now we have two translations and the function is:
.
g(x) = -x^2 - 5
.
We have one more translation to do. To shift the graph to the left 8 units, replace x
by x + 8. [Yep. The +8 shifts the graph to the left. If you had used -8, the graph would
shift to the right.] Anyhow, the resulting change to g(x) is:
.
g(x) = -(x + 8)^2 - 5
.
That's the answer. You might want to try plotting a few points of both functions just to
satisfy yourself of the shifts. Here's a graph of the two. The red graph is f(x) and the
green graph is g(x).
.

.
Hope this helps you out and that you can understand how the answer comes about.
.

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If you call the side of the original square x, then the area of the original square is x^2.
.
Now increase the length of each side by 6 units. So each side of the new square is x + 6.
This means that the area of this new square Bis (x + 6)^2 which multiplies out to x^2 + 12x + 36.
.
But the area new square is 4 times the area of the old square ... or 4x^2. So set the area
of the new square equal to 4 times the area of the old square. In equation form this is
.
x^2 + 12x + 36 = 4x^2
.
To put this into a more standard form, subtract 4x^2 from both sides to get:
.
-3x^2 + 12x + 36 = 0
.
Note that -3 is a factor of all terms on the left side. Therefore you can simplify
this equation by dividing both sides (all terms) by -3 to reduce the equation to:
.
x^2 - 4x - 12 = 0
.
Factor the left side to get:
.
(x - 6)(x + 2) = 0
.
Notice this equation is true if either of the two factors on the left side equals zero.
So set each factor equal to zero and solve for x.
.
x - 6 = 0 ... add 6 to both sides to get x = 6. This is one possible answer.
.
x + 2 = 0 ... subtract 2 from both sides to get x = -2 ... another possible answer.
.
But having a side of square equal to -2 doesn't make any sense. So the only reasonable answer
is that x ... the side of the original square ... is 6.
.
Let's check the answer. If the sides of the original square are 6 units, then the area of
the original square is 6 times 6 or 36 square units.
.
Increasing each side of the original square by 6 units makes the new square have sides of
12 units. Therefore, the area of the new square is 12 times 12 or 144 square units.
Notice that the area of the new square is 4 times the area of the original square ...
4 times 36 equals 144 square units.
.
Hope this helps you to understand the problem.
.

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An x-intercept is a point on the x-axis where the graph of the function crosses the x-axis.
But any (x,y) point on the x-axis has zero as its y-value. So to find the intercept
points on the x-axis, set y equal to zero and solve for the corresponding values of x. So
let's do that ... set y = 0 in the given function of . When you do that
you get:
.

.
Let's transpose this ... switch sides around ... to get it in the little more standard form
of:
.

.
Since the two terms on the left side both contain x, we can factor an x from each of the
terms to make the equation become:
.

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This equation will be true if either one of the factors equals zero because multiplying
the left side by zero will make it equal the right side which is zero.
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So the equation will be true if either x = 0 or x + 4 = 0. In the second factor this means
that x = -4. Since we already know that y is zero for these two values, the x-intercept
points are (0, 0) and (-4, 0).
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That's all there is to it.
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Hope this helps you to understand that to find x-intercepts you set y equal to zero and solve
for the corresponding values of x.
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Let T represent the tens digit. Let U represent the units digit.
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The problem first tells you that the tens digit is 3 more than the units digit. Write this
in equation form as:
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T = U + 3
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Next the problem tells you that if you square each of these digits and add the squares the
result is 117. The tens digit squared is T^2 and the units digit squared is U^2. Add these
two together and set this sum equal to 117. In equation form this is:
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T^2 + U^2 = 117
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But earlier we found that T = U + 3. So in the "squared" equation we can substitute
U + 3 for T to get:
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(U + 3)^2 + U^2 = 117
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Square the U + 3 term and the equation becomes:
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U^2 + 6U + 9 + U^2 = 117
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On the left side the two U^2 terms combine and the equation becomes:
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2U^2 + 6U + 9 = 117
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To get this equation into a form that can be solved, get rid of the 117 on the right side
by subtracting 117 from both sides to change the equation to:
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2U^2 + 6U - 108 = 0
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Simplify this a little by dividing both sides (all terms) by 2 to reduce the equation to:
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U^2 + 3U - 54 = 0
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Notice that this can be factored to:
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(U + 9)(U - 6) = 0
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This equation will be true if either of the factors is zero, because a multiplication
by zero on the left side will cause the entire left side to become zero and therefore equal
to the right side.
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Setting each of the factors equal to zero results in the possible solutions being U = -9
and U = +6. The minus solution makes no sense ... a minus digit in a number????
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So the solution is that the units digit is 6. And the tens digit is 3 more than that, so
the tens digit is 9. The number is 96.
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Check ... square the 9 to get 81 and square the 6 to get 36. The sum of 81 and 36 is 117.
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Everything works out, so our answer is correct.
.
Hope this helps you to understand the problem.
.

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You've given the wrong answers for this problem. The first clue is that the answers you
provided for Length and Width are in meters while the dimensions in your problem are in
centimeters. Another check is that the perimeter of the rectangle is supposed to be 18 cm.
However, your answers of Length = 16 and Width = 7 would give a perimeter of 16 + 7 + 16 + 7
which totals 46.
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So let's ignore your answer and work the problem using the information given.
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Begin by calling L the original length of the rectangle and calling W the original width.
The perimeter of the rectangle is the distance around it, and this is L + W + L + W which
combines to 2L + 2W. You are told that this perimeter equals 18 cm. So we can write the
equation:
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2L + 2W = 18
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and we can reduce this a little by dividing both sides (all terms) by 2 to get:
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L + W = 9
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Let's now solve this for one of the quantities in terms of the other. Let's say we solve
for L in terms of W. Do this by subtracting W from both sides of this equation to get:
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L = 9 - W
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One final step in this first part. Let's find the area of the original rectangle. We
know that the area (A) of a rectangle is found by multiplying the length L by the width W.
Therefore we can write the equation:
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A = L*W
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But we found that for this rectangle L is equal to 9 - W. Therefore, we can substitute
9 - W for L in the area equation and we get that the area of the original rectangle
is:
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A = L*W = (9 - W)*W = 9W - W^2
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Now to the second part of the problem. If we decrease the original length by 5 cm, the
new length is L - 5. Then if we increase the original width by 12 cm the new width is
W + 12. Therefore, the area of the new rectangle is found by multiplying the new length by
the new width which can be written as:
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New Area = (L - 5)(W + 12)
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But recall that L is equal to 9 - W, so replace L by 9 - W in the New Area equation to
get:
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New Area = (9 - W - 5)(W + 12)
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Combine the numbers 9 and -5 in the first set of parentheses and the equation reduces to:
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New Area = (4 - W)(W + 12)
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Multiply out the right side and the equation becomes:
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New Area = 4W + 48 - W^2 - 12W
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and combining like terms 4W and - 12W the equation reduces to:
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New Area = -W^2 - 8W + 48
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The problem tells you that this new area equals twice the old area. Recall that earlier we
found that the old area was 9W - W^2 so twice the old area would be 18W - 2W^2. Now we
can write the equation that the new area (-W^2 - 8W + 48) equals twice the old area (18W - 2W^2) ...
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-W^2 - 8W + 48 = 18W - 2W^2
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Let's get rid of the two terms on the right side. First get rid of the - 2W^2 by adding 2W^2
to both sides to get:
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W^2 - 8W + 48 = 18W
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Next get rid of the 18W on the right side by subtracting 18W from both sides:
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W^2 - 26W + 48 = 0
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Next factor the left side:
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(W - 24)(W - 2) = 0
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Note that this equation will be true if either of the factors on the left side is zero because
a multiplication by zero on the left side will make the left side zero and therefore
equal to the right side.
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By setting the first factor (W - 24) equal to zero and solving for W we get that W = +24 is
a possible answer. But is it really??? If the original width is 24, then the original
perimeter has to be much bigger than 18. So toss out the answer W = 24.
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By setting the second factor equal to zero we find that W = +2. That looks a lot better. Going
way back to the beginning of our work we had found that L = 9 - W and if W is 2, then
L = 9 - 2 = 7.
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So it looks as if the dimensions of the original rectangle is 7 cm long and 2 cm wide.
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Let's check. The perimeter of the original rectangle would be 7 + 2 + 7 + 2 = 18 cm. That's OK.
The Area of the original rectangle is 7 times 2 or 14 square cm.
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Now decrease the original length by 5 cm and you get 2 cm. Then increase the original
width by 12 cm and you get 2 + 12 = 14 cm. So the new figure has dimensions of 2 cm by 14 cm.
The area of this new figure is the product of 2 cm by 14 cm which equals 28 square cm.
And this area (28) is twice the square cm of the original rectangle (14). So everything
works out and our answers of 2 cm for the original width and 7 cm for the original
length are correct.
.
A lot of work ... hope this helps you to see your way through this problem.
.

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Yes, it is a perfect square
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Since the coefficient (multiplier) of the x^2 term is 1, the first term can only be factored
to x times x. This means the two factors of x^2 + 18x + 81 must be of the form:
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(x + ____)*(x + ____)
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The blanks must be a factor pair of 81. The only pairs of factors of 81 are 81*1 and 9*9.
If you fill the blanks of the factors ... for checking ... with the pair of 9's you get:
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(x + 9)*(x + 9)
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If you multiply these out you will get x^2 + 18x + 81.
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But note that (x + 9)*(x + 9) = (x + 9)^2
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So the trinomial is equal to (x + 9)^2 and is, therefore, a perfect square.
.
Hope this helps you to understand the problem.
.

You can put this solution on YOUR website!
Let one of the integers be x and the other integer by y
.
The product of the integers is 98. That means that x times y equals 98 and in equation form
this is:
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x*y = 98
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The integers differ by 7. This means that if we subtract 7 from one of the integers it
will equal the other integer. So we can say that:
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x - 7 = y
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This tells us that y = x - 7 so in the first equation we can substitute x - 7 for y. This
makes that product equation become:
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x*(x - 7) = 98
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Multiply out the left side to get:
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x^2 - 7x = 98
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Subtract 98 from both sides of this equation and you get:
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x^2 - 7x - 98 = 0
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This equation can be factored to give:
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(x - 14)(x + 7) = 0
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This equation will be true if either of the two factors equals zero ... because multiplication
by a zero on the left side makes the left side equal the zero on the right side.
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So set the two factors equal to zero and solve for the value of x that makes each factor
equal zero:
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x - 14 = 0
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Add 14 to both sides and you get x = +14
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Then do the second factor:
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x + 7 = 0
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Subtract 7 from both sides to get x = -7
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But x can't be -7 because the problem says the integers are positive. Therefore, the
only valid solution for x is that x = +14.
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Now we can go back to the first equation ... the equation says that the product of the two
integers is 98:
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x*y = 98
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But x is +14. Substituting this into the equation results in:
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14*y = 98
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Divide both sides of this equation by 14 to solve for y and you get:
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y = 98/14 = +7
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So the two integers (x and y) are 14 and 7. Their product is 98 and their difference
is 14 - 7 = 7 ... just as the problem required.
.
Hope this helps you to understand the problem and how to solve it.
.

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Try this ID number:
.
381654729
.
Hope this helps.