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When you read the problem, it becomes apparent that you need to find a small number and a
larger number. For ease of understanding and identification, call the Small number S and the
Larger number L.
.
The first sentence tells you that one number is 8 more than another. If it is 8 more, then
this means that it is L. So you should get the idea that if you take 8 away from L then you
should have S. You can write this as:
.
L - 8 = S
.
or by adding 8 to both sides, it may be clearer to you as S + 8 = L. Either one is OK to use.
.
The second sentence tells you that the sum of the Small number (S) and twice the Larger number
(2*L) equals 46. Writing this in equation form gives you:
.
S + 2L = 46 <- call this the second equation
.
Then, from the first equations you can either use the fact that S = L - 8 or that L = S + 8
and substitute into the second equation. For ease, let's use S = L - 8 and so we can replace
S in the second equation with L - 8 to get:
.
L - 8 + 2L = 46
.
Next you can get rid of the -8 on the left side by adding 8 to both sides. When you do
this addition you get:
.
L + 2L = 54
.
Add the terms on the left side and the equation reduces to:
.
3L = 54
.
Finally divide both sides by 3 to reduce the equation and solve for L:
.
L = 54/3 = 18
.
Now you know that L = 18 you can solve for S. You know that L is 8 larger than the smaller
number S. So you can subtract 8 from L and you have S. Therefore, L - 8 = 18 - 8 = 10. Therefore,
S is 10.
.
So the two numbers are 10 and 18.
.
Obviously the larger number is 8 more than the smaller number. Then check further by
adding the smaller number to twice the larger. If you do you get 10 + 2*18 and this becomes
10 + 36 and this equals 46, just as the problem says it should.
.
Hope this helps you to understand the problem and how to solve it.
.

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Something is wrong here.
.
The distance between the two points is found by using the Pythagorean theorem. The change in
x values gives you one leg of the triangle, and the change in y values gives you the other
leg.
.
Let's compute the change in x. In one of the points the x-value is -3. In the other point
the x-value is +1. So x goes from -3 to +1 and that is a distance of 4 units.
.
Now let's compute the change in y. In one of the points the y-value is -2, and in the other
point it is +4. So y goes from -2 to +4 and that is a distance of 6 units.
.
So we know that the two legs of the right triangle are 4 and 6. The hypotenuse of the
triangle is the distance between the points. The Pythagorean theorem tells us that the
sum of the squares of the two legs is the hypotenuse squared. In equation form this is:
.

.
where a is one of the legs, b is the other leg, and c is the distance we are looking for.
.
Let's call a the x distance of 4 and b the y distance of 6. Substituting these values we
get:
.

.
and squaring out the terms on the left side results in:
.

.
Adding the terms on the left side simplifies the equation to:
.

.
Now solve for c by taking the square root of both sides to get:
.

.
If you use a calculator to find you get .
So the distance between the two points you were given is 7.211102551 units.
.
But none of your answers matches that value.
.
Answer a) is square root of 10 and in your head you can tell that it has to be a little bigger
than square root of 9 which is 3 but it is less than the square root of 16 which is 4.
Therefore, answer a) is between 3 and 4, which means that it can't match our answer of
7.211102551.
.
Answer b) is 27. That definitely doesn't match our answer.
.
Answer c) is and you know that is less than
which is 3. So must be less than and that means that answer c) is
less than 6. Therefore, it doesn't match our answer.
.
Finally, Answer d) is . You know that is greater than 1. If
it were just 1, then this answer would be . So this answer is greater than
18. Therefore, it can't be our answer of 7.211102551.
.
None of the answers matches.
.
Like I said at the beginning ... something is wrong here with this problem.
.
Hope this helps you to see your way through the problem. The answer you are trying to
match (assuming that the points are correct) is .

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The volume of a cylinder is found by multiplying the area of the circular base times the
height. And the area of the circular base can be written two ways depending on whether
you choose the diameter or the radius to work with. Let A = area, R = radius, and D = diameter.
Then the area of a circle is either:
.

.
or by recognizing that the radius is half the diameter and then substituting
for R in the above Area equation you get:
.

.
Next the Volume (V) of the cylinder is just the height (H) times A or H*A. Substituting
for A we can write:
.

.
or
.

.
Now all you have to do is to plug in the values that you are given for V and H and solve
for either R or D. Let's work a couple of examples. (Make sure you use a consistent
set of units. If the height is in inches, then you need the volume in cubic inches
and the diameter and radius will be in inches also. The area will be square inches.)
.
Suppose you are given that the volume is 400 cubic inches (V = 400) and the height is
10 inches. To find the radius, use the equation:
.

.
Substitute 400 for V and 10 for H and you get:
.

.
Now divide both sides by the multipliers of and you get:
.

.
and solve for R by taking the square root of both sides:
.

.
If you use your calculator to find you find that
and the square root of that is . Substituting this into the equation
gives you:
.

.
and since we are working in inches, our answer is in inches also.
.
You can find the diameter by just multiplying the answer for R by 2 to get 7.136496465
inches or you can use the equation that we got above involving the volume and the diameter.
Let's use the equation:
.

.
Substitute 400 for V and 10 for H to get:
.

.
Multiply both sides of this equation by 4 to get rid of the denominator on the right side.
When you do that, the equation becomes:
.

.
Divide both sides by and the equation simplifies to:
.

.
Use a calculator to determine that . This reduces the
equation to:
.

.
Finally, by taking the square root of both sides, you get:
.

.
and 7.136496465 inches for the diameter is the same answer that we got above by doubling
the radius, so it checks.
.
Hope this helps you. If you study the problem, as complex as it seems, all you have to
know is the formulas for the area of a circle and the fact that the volume of a cylinder is
found by multiplying the circular area of the cylinder times the height of the cylinder.
The rest of the problem just consists of plugging in the values you are given and solving
the resulting equation.

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I'm not sure whether you meant the problem to be find f(-2) given that
.
<=== first case
.
or
.
<=== second case
.
In either case, the way you work this problem is to substitute -2 for x and calculate the
answer.
.
In the first case, substituting -2 for x results in:
.

.
In the second case, substituting -2 for x results in:
.

.
[Note that in these two problems and ]
.
Hope this helps you to see your way through the problem. Just remember that when you
write f(-2) it means substitute -2 for the variable on the right side. So f(5) would
mean substitute +5 for the variable, and so forth.

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Given:
.

.
Note that this product is covered by the rule for the product of the sum and difference
of two quantities. This rule is:
.

.
If you actually do the multiplication of (a - b) times (a + b) you will see why this rule
is true.
.
Anyhow, if you let and then, according to the rule, the product
will be or .
.
But squaring the square root of 2 results in an answer of 2 and squaring 3 results in 9.
.
So according to the rule, the answer is:
.

.
The answer is -7 and notice that the radicals are gone completely.
.

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I suspect that you meant to say the square root of 50 instead of the square of 50. So you
meant for the problem to be:
.

.
Note that 50 is 25*2. So you can substitute 25*2 for 50 and the problem becomes:
.

.
By the rules of square roots you can split into
to convert the problem to:
.

.
But the square root of 25 is 5 and substituting 5 for the square root of 25 changes the
problem to:
.

.
Then you can factor out the square root of 2 to make the problem:
.

.
and replace 3+5 by 8 makes it:
.

.
and we conventionally write this as:
.

.
Hope this helps you to understand the problem and some of the steps you can take to simplify
radicals.
.
If you really meant the problem to be:
.

.
then please post the problem again because that is a whole different problem from the
one I worked above.

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You need to work in consistent units. Note that the weight of the bottle is given in grams
and the combined weight of the bottle and water is given in kilograms. Convert the weight
of the combined water & bottle from kilograms to grams. Do this by multiplying 1000 times
the weight in kilograms to find that the combination weighs:
.
1.22 kg * 1000 = 1220 grams
.
Next you can subtract the weight of the bottle (115 grams) from the weight of the bottle-water
combination to find the weight of only the water as follows:
.
Weight of Water = 1220 gms - 115 gms = 1105 gms
.
And since each gram is 1 ml of water, there must be 1105 ml of water.
.
Hope this helps you to see your way through this problem.
.

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The problem tells you that each 1/4 inch on the drawing equals 12 inches in "real world"
dimensions. And since 12 inches is 1 foot, you can say that each 1/4 inch on the drawing
represents 1 foot in actual size.
.
In the explanations below, you will see 3 3/8 written as 3+3/8 in some places. These are
equivalent ways of writing the mixed number.
.
If you measure 3 3/8 inches on the drawing, how many 1/4 inches is that? Since each inch
contains 4 quarter inches, the three inches is 4*3 or 12 quarter inches. You still have the
3/8 inches, and 3/8 of an inch contains one 1/4 plus half of another or 1.5 quarters.
So the total number of 1/4 inches is 12 + 1.5 = 13.5 quarters and since each of those
quarters represents a foot, the total number of feet is 13.5 feet.
.
If this explanation is a little confusing, there are a couple of other ways you can look
at it. You can find the number of quarters in 3 3/8 by dividing 3 3/8 by 1/4. Convert
the 3 3/8 to an improper fraction by multiply 8 times the whole number 3 and adding
the additional 3 from the numerator to get 3 3/8 = (24+3)/8 = 27/8. To divide this by
1/4, invert the 1/4 to get 4/1 and multiply that by the 27/8. The result is:
.
(27*4)/(8*1) = 118/8 = 13 4/8 = 13 1/2. This again tells you that the drawing measurement
of 3 3/8 contains 13 1/2 quarter inches and since each quarter inch represents 1 foot,
the drawing dimension represents 13 1/2 feet.
.
Or you could set up the proportion of one-quarter inch is to 1 foot as 3 3/8 inches
is to X feet which translates to the equation:
.

.
Solve this equation by cross multiplying ... multiplying the numerator of the left side
times the denominator of the right side and setting that product equal to the numerator
of the right side times the denominator of the left side. When you do that you get:
.

.
If you multiply both sides of this equation by 4 to eliminate the denominator on the left
side, you get:
.

.
The distributed multiplication on the right side leads to:
.

.
Again the answer is 13 1/2 feet.
.
Hope one or more of these methods makes sense to you.

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Let F represent the current age of Henry's father. And Let H represent Henry's current age.
.
Currently the father is twice as old as Henry. Therefore, if you double Henry's current
age, the result equals the father's age. In equation form this is:
.
F = 2*H
.
Sixteen years ago, the father's age was his current age less 16 which is F - 16. The same
with Henry. Sixteen years ago Henry was his current age less 16 or H - 16. And at that time
the father's age (F - 16) was equal to 4 times Henry's age [4 times (H - 16)]. In equation
form this is:
.
F - 16 = 4*(H - 16)
.
Simplify this by multiplying out the right side to get:
.
F - 16 = 4H - 64
.
Then get rid of the -16 on the left side by adding 16 to both sides. When you add 16
to both sides you get:
.
F = 4H - 48
.
Now from the first equation we wrote, F = 2H. So we can substitute 2H in place of F
in the equation and with this substitution the equation
.
F = 4H - 48
.
becomes:
.
2H = 4H - 48
.
Next let's get rid of the 4H on the right side by subtracting 4H from both sides.
When you subtract 4H from both sides, the equation becomes:
.
-2H = -48
.
Divide both sides by -2 which is the multiplier of H and you get:
.
H = -48/-2 = 24
.
This tells us that Henry is currently 24 years old. And since we know that his father
is presently twice as old as he is, Henry's father must currently be 48 years old.
.
Let's check that. Sixteen years ago, Henry's father was 48 - 16 = 32 years old and also
16 years ago, Henry was 24 - 16 = 8 years old. So 16 years ago the father's age (32) was
4 times Henry's age (8). That works! So the answer to the problem is that Henry is currently
24 years old and his father is currently 48 years old.
.
Hope this helps you to understand the problem and a process that you can use to solve it.

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If you are saying that the answer is:
.

.
You are correct.

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Given:
.
-kx - 4y= -k
.
This equation must be true for the point (-2, 3) which means it must be hold true when
x is -2 and y is +3.
.
Substitute those values in for x and y to get:
.
-k(-2) -4(+3) = -k
.
Multiply out the two terms on the left to get:
.
+2k - 12 = -k
.
Next, we need to get the terms containing k on one side of the equation and everything
else on the other side. We can get rid of the -k on the right side by adding +k to both
sides. On the right side the addition of -k and +k results in zero. And on the left side
the +k adds to the +2k. This results in:
.
3k - 12 = 0
.
Next add 12 to both sides to get rid of the -12 on the left side. With this addition
the equation becomes:
.
3k = 12
.
Solve for k by dividing both sides by 3 ... the multiplier of k. The result of this
division give us the value of k as:
.
k = 12/3 = 4
.
check the answer by substituting 4 for k in the original given equation. When you do,
the equation:
.
-kx - 4y= -k
.
becomes:
.
-4x -4y = -4
.
Divide all terms by -4 to get:
.
x + y = 1
.
Does (-2, 3) satisfy this equation? Substitute -2 for x and +3 for y to get:
.
-2 + 3 = 1
.
Add the terms on the left side and you get:
.
1 = 1
.
This verifies that k = 4 because it works in the equation and (-2, 3) is a solution
when k has this value.
.
Hope this helps you to understand the problem.

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George is 12 years older than Jeannie. The sum of their ages in 5 years will be 50. How old is Jeannie now?
a. Define the variables.
.
Just to help us remember what is what, let's use G to represent George's current age and
J to represent Jeannie's current age.
.
b. Write an expression for the age of George in terms of Jeannie's age.
.
The problem tells you that George is 12 years older than Jeannie. That means if you subtract
12 from George's age, the answer should be Jeannie's age. In equation form this would be:
.
G - 12 = J
.
Solve for G by adding 12 to both sides to get:
.
G = J + 12
.
c. Write an equation for the sum of their ages in 5 years from now.
.
Five years from now George's age will be G + 5 and Jeannie's age will be J + 5. The problem
tells you that the sum of these ages will be 50. In equation form this would be:
.
G + 5 + J + 5 = 50
.
By adding the two 5s on the left side this simplifies to:
.
G + J + 10 = 50
.
And subtracting 10 from both sides further simplifies this equation to:
.
G + J = 40
.
d. Solve the equation for the age of Jeannie now.
.
Start with the equation from part c:
.
G + J = 40
.
But way back in part b it is known that G = J + 12. Substitute the right side of this
equation in place of G and get:
.
J + 12 + J = 40
.
Add the two Js and the equation becomes:
.
2J + 12 = 40
.
Subtract 12 from both sides results in:
.
2J = 28
.
Divide both sides by 2 to solve for J and you get:
.
J = 28/2 = 14
.
This tells us that Jeannie is 14 years old at present.
.
That means that George, who is 12 years older, is 26 at present.
.
Five years from now, Jeannie will be 14 + 5 or 19 years old. And George will be 26 + 5
or 31 years old. The sum of their ages will be 19 + 31 which is 50, just as the problem
said it should be.
.
Hope this helps you to understand the problem.

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Given:
.
M = 0.85(220 - a)
.
and also given that M = 133
.
Find "a"
.
Method:
.
Substitute 133 for M in the equation to get:
.
133 = 0.85(220 - a)
.
Multiply 0.85 times each of the quantities in the parentheses:
.
133 = 0.85*220 - 0.85*a
.
Note that 0.85 times 220 is 187. When you do that multiplication, the equation becomes:
.
133 = 187 - 0.85a
.
Next you need to get the term containing "a" by itself on one side of the equation, and
everything else on the other side. Let's begin by adding 0.85a to BOTH sides of the equation.
When you do, the addition of 0.85a to the right side cancels the -0.85a on the right side.
This makes the equation become:
.
133 + 0.85a = 187
.
Subtract 133 from both sides to get rid of it on the left side. When you subtract
133 the equation becomes:
.
0.85a = 187 - 133
.
The subtraction on the right side results in:
.
0.85a = 54
.
Divide both sides by the multiplier of "a" to solve for "a". When you do that the equation
becomes:
.
a = 54/0.85
.
and the right side is:
.
a = 63.5294
.
which tells you that at the age of 63.5 years the maximum rate of heart exercise is 133.
.
You can check this by substituting 63.5 into the original equation to find that:
.
M = 0.85(220 - 63.5)
.
Subtract the two numbers in parentheses to get:
.
M = 0.85(156.5)
.
do the multiplication and the result is:
.
M = 133.025
.
The slight difference is due to rounding off the number of years. So the answer we got in
this checking work is equal to the heart rate specified by the problem ... 133.
.
Hope this helps you to understand how to work the problem.

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Given:
.
y>x+1 and y<3/2x+3
.
Problem: Solve and Graph
.
First y > x +1
.
Let's begin by treating this as an equality y = x +1. We will graph this. What can you tell
by looking at this equation. To begin with, this is a first degree equation (x has an
implied exponent of 1, not 2, 3, or ...). First degree equations have straight lines as
graphs. Next, note that this is in the slope-intercept form y = mx + b in which m (the multiplier
of x) is the slope of the graphed line (m = 1) and +1 is the point on the y-axis where the
graph crosses. Furthermore, you can graph the solutions of this equation by assuming a
couple of values for x and finding the corresponding values of y. For example, let x equal
zero. The equation then becomes y = 0 + 1 which simplifies to y = 1. So you know the point
(0,1) is on the graph. Next let x = -1. The equation becomes y = -1 + 1 = 0. So the point
(-1,0) is on the graph. Next let x = 2. The equation becomes y = 2 + 1 = 3. So the point
(2, 3) is also on the graph. You can continue this process and find a few more points.
Plot all the points and use a straight edge to draw a line through all the points you
have plotted. What does this line represent? It consists of all the points in which
y EQUALS x + 1. But the problem says that y must be GREATER than x + 1. Therefore,
any point that is above this graphed line has a y value that is greater than x + 1. The
way this is normally indicated is by graphing the line for the equation y = x + 1 and
then shading the area ABOVE the line (but not including the line itself). Any point in
the shaded are then has a y value that is more than the value of its corresponding
value of x plus 1. For example, if you let x = 5 then x + 1 is 6. So any point that has
an x value of 5 and a y value of greater than 6 satisfies y > x + 1. Examples of such
points are (5, 7), (5, 8), (5, 9), (5, 6.00001). If you plot these points you will see
that they are in the shaded area above the graph line for the equation y = x + 1.
.
Here's the graph:
.

.
Imagine the shaded area above the line as the solution to this problem.
.
Next, let's work the problem:
.

.
Same approach as before. Change this to an equation:
.

.
Graph it. Your graph should look like this:
.

.
This time though since y is LESS than the points having this property
will be BELOW the line. Shade in the area below the line (do not include the line in
the shading).
.
Some points in the shaded area are (2, 5), (2, 4), (4, 8), (4, 6), (0, 2), (0, 0), (-2, -1),
(-2, -3). For each point, go to the given inequality, plug in the value for x and its
corresponding value for y. See if the inequality holds. That tells you that the point
satisfies the inequality. As an example, pick the point (2, 5). Plug 2 in for x and 5 in
for y in the inequality:
.

.
Substitution results in:
.

.
Multiply on the right side to get:
.

.
Note this inequality is true ... so (2, 5) is a point that satisfies the inequality
and also note that the point is in the shaded area.
.
One more point that does not apply to this problem ... if the inequality has an equal
sign as part of it, the graphed line IS INCLUDED in the shaded area. For example, if
the original inequality for the first problem had been:
.

.
the inclusion of the equal sign with the inequality sign would tell you that points on
the graphed line would be solutions also. Therefore, the line would need to be included
in the shaded region.
.
Hope this helps you ...

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Anytime you see an equation that has a variable in the exponent, one of the first things
you should think of is "logarithms"
.
You are given:
.

.
Take the logarithm of both sides of this equation. You can use base 10 or base e, which is
"ln" or natural logarithms. These can be worked on a normal scientific calculator so either
is a good choice. Let's choose base 10. Taking the Log base 10 of both sides of the equation
results in:
.

.
To simplify things a little, let's use a calculator to find . Enter 625
on the calculator and press the "log" key. You should get 2.795880017. Substituting
this value results in the equation becoming:
.

.
Next we'll use another property of logarithms. If you are finding the logarithm of a quantity
that has an exponent, an equivalent form is to multiply the logarithm by the exponent.
In other words, is equivalent to so let's substitute
that into our equation to get:
.

.
Now use a calculator to find . Enter 125 and press the "log" key to get
. Substitute this into the equation and you now have:
.

.
Finally divide both sides by 2.096910013 and the answer becomes:
.
and this is
.
Hope this helps you with your understanding of logarithms.

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By "In" I believe you mean "ln" which is the notation for natural logarithms. Natural
logarithms have the base e which is the symbol for a number that is approximately 2.718281828 ...
.
This is a problem in which you can use a translation from logarithmic form to the equivalent
exponential form. This translation is done according to the following rule:
.
is equivalent to the exponential form
.
In your problem, the "ln" tells you that you are using a logarithm with the base e. Therefore,
your problem can be written:
.

.
and by comparing this to the translation rule you can tell that b = e, C = 3, and A = x.
Substitute these values into the exponential form and you get an exponential form of:
.

.
So you now know that and since e is approximately 2.718281828 you can get
an approximate value for x by using the equation:
.

.
You can then use a calculator to cube 2.718281828 and you will get:
.

.
as a pretty close approximation of the value of x.
.
Remember that translation form. It's a fundamental definition for logarithms and it
comes in quite handy for solving lots of logarithmic problems.
.

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Given:
.

.
Solve for x
.
You can convert this problem to exponential form to solve it. The conversion between
logarithmic and exponential form is done according to the following relationship:
.
is equivalent to the exponential form
.
Remember this relationship. It's quite useful in solving logarithmic problems.
.
If you compare what you are given to the log form you can see that b = 5, C = 30, and A = x.
.
Now substitute these values into the exponential form and you get:
.

.
So and that is a pretty big number. You can use a calculator to find that
is approximately
.
These are answers that you are looking for. Let's work a similar problem with smaller numbers
just for practice. Suppose the problem had been:
.

.
You solve for x by converting this log form to exponential form. In this case b = 5,
C = 3, and A = x. Substituting these values into the exponential form results in:
.

.
and 5 cubed is 5*5*5 = 125. So for this problem, x would be 125, and you can now write:
.

.
Hope this helps you to understand the problem a little better
.

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Given:
.

.
The first thing you can do is to divide the 3 in the denominator into the 9 in the numerator.
The result of this division in that you are left with a 3 in the numerator. [You can also
replace the 9 with 3*3 and then cancel the 3 in the denominator with one of the 3s in the
numerator. The result is the same ... a 3 left in the numerator.] The problem is now simplified
to:
.

.
Next you can cancel the (x-7) in the denominator with the (x-7) in the numerator.
This cancellation is as follows:
.

.
And the simplification asked for by the problem is what is now leftover, and is:
.

.
If you want to, you can perform the distributed multiplication by 3 to get:
.

.
Either answer [3(x + 7) or 3x + 21] is correct.
.
Hope this helps you to understand the problem.
.

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Divide the following:
.

.
You can split this into two terms as follows:
.

.
The term of 18 divided by 9 is equal to 2. This simplifies the problem to:
.

.
Note that the denominator of 9 is equal to . If we replace the 9 with this
radical the problem now becomes:
.

.
By the rules of radicals we can put both terms under the same radical sign to convert
the problem to:
.

.
and note that 81 goes into 567 exactly 7 times. So replace 567/81 by 7 and the problem
simplifies to:
.

.
Let's check with a calculator. First, the original problem:
.
Square root of 567 = 23.8117618
.
Add 18 to this and you get 41.8117618
.
Divide this total by 9 and the result is 4.645751311
.
Next let's use a calculator to check our answer.
.
The square root of 7 is 2.645751311 and if you add 2 to that you get 4.645751311.
.
The two answers agree, so we have a correct solution to the problem. The original
problem simplifies to
.
Hope this helps you to understand the original problem a little better.
.

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Here's an approach to this problem.
.
If the number of policies sold is 10,000, the probability is that of this pool of policies
0.0002 is the probability of death. Multiply the 10,000 by 0.0002 and you get that 2 deaths
will occur.
.
At $25 per policy, 10,000 policies will bring in a total of $250,000 (which comes from
$25 * 10,000).
.
But the company will statistically have to pay for two policies, and since the payoff
will be $20,000 per policy for a total payoff of $40,000.
.
Therefore, the "profit" on $250,000 of income to the company will be $250,000 less the
$40,000 payoff which is $210,000.
.
Dividing this profit by the 10,000 policies that generate it will give the "per policy
profit".
.

.
Therefore, you can say that of the $25 the company collects for a policy, the policy
will typically return $21 as "profit".
.
Hope this helps you to understand the problem and a way that you can analyze it to get
an answer.

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How about doing it this way:
.

.
Then by the rules of square roots, you can use the rule that:
.

.
So you can write:
.

.
But
.
Substituting 6 for in the right side of the simplification equation you get:
.

.
So the answer to this problem is simplifies to
.
Hope this helps you understand the simplification process for radicals.

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Since 500 is an even number you know that it is divisible by 2. Therefore, you can write
it as
.
But 250 is also even and divisible by 2. So 250 can be replaced by 125 * 2. Therefore,
you can replace 500 by 125*2*2 which is equivalent to
.
Then . So if you substitute for 125 you get:
.

.
Taking the square root of all three terms in this multiplication string results in:
.

.
So the simplified form of is
.
Had you recognized it earlier, you could have written:
.

.
much quicker, but not as much practice as the long way.
.
Hope this helps you to see how square root problems such as this one can be simplified.
.

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An easy way to find the slope is to work the equation into the slope-intercept form which
is given by the equation:
.
y = mx + b
.
In this form m is the slope and b is the value on the y-axis where the graph crosses
the axis.
.
So take the given equation 3x + 2y = 6 and solve it for y.
.
First step, subtract 3x from both sides to get:
.
2y = -3x + 6
.
Then divide both sides (all terms) by 2 to solve for y. When you do that division you
get:
.
y = (-3/2)x + 3
.
By comparing this with the slope intercept form, you can see that m, the slope, is -3/2
and you can tell that the graph crosses the y-axis at the value y = +3.
.
Hope this helps you understand the problem.

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The form of the equation that you need for this problem is called the slope-intercept
form. It is given by the equation:
.
y = mx + b
.
where m is the slope of the graph of the equation and b is the y-intercept, that is, the
value on the y-axis where the graph crosses.
.
The first part of the problem gives you the equation:
.
2x - 3y = 6
.
and asks you to find the slope and the y-intercept of the graph. The easiest way to do that
is to rearrange the equation into the slope intercept form. Once you do that you know that
the slope (m) is the multiplier of the x and the constant is the y-intercept
.
So we need to solve the given equation for y. The first step is to isolate the y term on
one side of the equation and have everything else on the other side. So let's begin by
subtracting 2x from both sides of the equation. When you do that the equation becomes:
.
-3y = -2x + 6
.
Next solve for y by dividing all terms on both sides of the equation by -3, the multiplier
of the y term. When you do that division the equation becomes:
.
-3y/3 = -2x/-3 + 6/-3
.
and this simplifies to:
.
y = (2/3)*x - 2
.
Now by comparing this equation with the slope-intercept form you can tell that the slope
(the multiplier of x represented by m) is (2/3) and the y-intercept (the constant
represented by b) is -2. Those are the answers to the first part of this problem.
.
The remainder of the problem can be done just by substituting values into the slope-intercept
equation form.
.
The first of the three givens tells you that the slope is 5 and the y-intercept is (0, -2).
The intercept point on the y-axis is, therefore, -2. The slope intercept equation form
becomes:
.
y = 5x - 2
.
You can graph this by marking the point -2 on the y-axis. You know that the graph goes
through this point. Put your pencil on that point. Move in horizontally to the right
1 unit. Stop. Then move it vertically upward 5 units. Stop and mark that point. This is
a second point on the graph. Now you can run a straight line through the two points and
you have the graph. Another way you can find points is to just assume values for x,
plug them into the equation, and solve for y. The point (x, y) will then be on the graph.
For example, assume x = 2. Then the equation for y tells you that y = 5(2)-2 = 10 - 2 = 8.
So the point (2,8) is on the graph. When you get done, the graph should look like:
.

.
You can do the remaining two problems the same way. For the second one you are given that
the slope is -2 and the intercept is (0,4). That means the slope intercept equation is:
.
y = -2x + 4
.
Since the graph crosses the y axis at +4 you can mark that point on the y-axis. Start your
pencil on that location. Go horizontally to the right 1 unit and at that location go
down -2 units because the slope is -2. (You should be at (1,2)). That is a second
point on the graph. Since you now have two points on the graph you can draw a line through
these points and that is the graph you need. Again you can also find additional points by
assuming values for x and solving for corresponding values of y. Then plot the (x,y) points.
When you get done the graph should look like:
.

.
The final problem gives you a slope of -3/4 and a y-intercept of 8. The equation for this
is:
.
y = -(3/4)x + 8
.
Mark +8 on the y-axis. Then go horizontally to the right from that point 1 unit followed
by going vertically down 3/4 of a unit. Mark that point. (or you can go horizontally
to the right 4 units from the point on the y-axis and then go vertically down 3 units).
You can also substitute values for x into the equation and find corresponding values
of y to give you more points on the graph. When you get done, your graph should look like:
.

.
Hope this helps you to understand the slope-intercept form a little better.

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Force is defined as mass times acceleration. And in SI units, mass is in units of kilograms and
a generally accepted figure for the force of gravity is 9.8 meters per second^2.
.
In this problem, the total mass acting on the wire is 70 + 82 + 20 + 17 = 189 kilograms.
.
Substituting this and the acceleration due to gravity into the Force equation results in:
.
F = m*a = 189 * 9.8 = 1852.2 kg/m/sec^2
.
and since a Newton is equal to a kg/m/sec^2, the answer is
.
1852.2 Newtons
.
Hope this helps you to understand the relationship between mass (kilograms) and Force (Newtons).

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Given:
.
divided by
.
To divide by a fraction you can invert that divisor and multiply it times the dividend.
Therefore, this division problem converts to the multiplication of the dividend by
the inverted divisor which is now ...
.

.
Just multiply the terms in the two numerators to get the resulting numerator, and also
multiply the terms in the two denominators to get the resulting denominator. This makes
the problem become:
.

.
So the correct answer is B ...
.

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Given:
.

.
Divide it by:
.

.
Recall the "arithmetic" rule about fractions. If you are dividing by a fraction you can
invert the divisor and multiply it by the dividend. In this case it means you can take
and multiply it by the inversion of to get the answer
to this problem. After the inversion, the original division problem is converted to the
following multiplication problem:
.

.
Before it gets too complicated, let's do some factoring. Note that:
.
factors to and
factors to and
factors to and
factors to
.
Substitute all these factored forms into the appropriate places in the multiplication
problem and it becomes:
.

.
Cancel all the terms that appear in the numerator with a corresponding appearance in
the denominator as follows:
.

.
The remaining factors in the numerator are 4 and 2. They multiply out to 8. And the remaining
factors in the denominator are 7 and 5 and they multiply out to 35. This makes the answer:
.
and there are no more common factors in the numerator and the denominator.
So this cannot be reduced further and the answer to the original problem is
.
Hope this helps you to understand the division process in this problem.
.

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Given:
.

.
Subtract
.
Notice first that factors into . Substitute the factored
form into the problem, and the problem becomes:
.

.
Next, create a common denominator in the two terms by multiplying the first term by
which is equivalent to multiplying by 1 because the numerator of this
multiplier is the same as the denominator. This multiplication makes the problem
become:
.

.
and this simplifies to:
.

.
The numerator of the first term multiplies out to . Substituting this into
the polynomial makes it become:
.

.
Now since the two terms have the same denominator, they can be combined by just subtracting
the two numerators and putting that result over the common denominator. This results
in:
.

.
Now notice that the numerator factors into .
Substitute this for the numerator and the result is:
.

.
Cancel the like terms in the numerator and denominator:
.

.
and this reduces to the form:
.

.
This is the answer you are looking for. Hope this helps you to understand the problem
and how to work it through to an answer.
.

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Given:
.

.
Find y
.
First you can do . Substitute this and the equation becomes:
.

.
Now start with the most nested (the interior) set of parentheses. These are the parentheses
that contain 2 - 3. Since 2 - 3 = -1 we can substitute this and the equation becomes:
.

.
Then multiply 7 times the -1 and get -7. Substitute -7 for 7(-1) and the equation then
becomes:
.

.
Then subtract 7 from 8 to get +1. Substitute +1 for the parentheses and you have:
.

.
Multiply 3 times +1 and the equation is reduced to:
.

.
and the answer (y) is equal to 12 + 3 and this is +15
.
Hope this explains where the answer came from.

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Given:
.
5x + 41 = -4
.
The problem asks you to solve for x.
.
The first basic rule is whatever you do to one side of this equation (add, subtract,
multiply, divide, square, square root ... and so on), you must also do to the other side
or else you "upset" the balance of the equation.
.
The process of solving equations like this one involves four steps ...
.
(1) get the term or terms containing the unknown alone on one side of the equal sign.
(2) get all the number terms on the other side of the equal sign
(3) combine the like terms on their sides
(4) divide by the multiplier of x
.
Let's apply the process one step at a time:
.
First we need to get all the terms containing "x" on one side. Hmmm. There is only one
term containing "x" and it's on the left side. So we're OK there.
.
Next we need to get rid of the +41 on the left side so the "x" term is left by itself on
the left side. Do that by subtracting 41 from the left side. But when you do that, don't
forget the rule. If you subtract 41 from the left side, you must also subtract 41 from
the right side. This subtraction is:
.
5x + 41 - 41 = -4 - 41
.
On the left side the +41 and the -41 cancel each other. As a result, the equation becomes:
.
5x = -4 - 41
.
On the right side the -4 and the -41 are like terms that combine to give -45. So the equation
is now reduced to:
.
5x = -45
.
Now you do the last step. You see that 5 times the unknown equals - 45. To find 1 times the
unknown (1x or just x) you divide both sides by the multiplier of x which is 5. Dividing
both sides by 5 (remember the rule ... divide one side by 5 and you must also divide the
other side by 5) and you get:
.
5x/5 = -45/5
.
and this simplifies to just:
.
x = -9
.
You can check this answer by going back to the original given equation and substituting
-9 for x to see if the equation balances. The original equation was:
.
5x + 41 = -4
.
Substitute -9 for x and you get:
.
5*(-9) + 41 = - 4
.
Do the multiplication 5*(-45) and the equation becomes:
.
-45 + 41 = -4
.
The terms on the left side (-45 and + 41) combine to give you -4 and this makes the equation:
.
-4 = -4
.
And since both sides are equal, the equation balances and the answer of x = -9 is correct.
.
Hope this helps you with your understanding of how to do equations of this sort.

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Let L represent the largest number
Let M represent the middle number and
Let S represent the smallest number
.
If you were to roll over the numbers one at a time until all three numbers show, there are
six possible outcomes:
.
L then S then M
L then M then S
M then S then L
M then L then S
S then M then L
S then L then M
.
Since all of these are equally likely, they each have a probability of .
.
Notice that two of the outcomes have L as the first draw. That means that the possibility
of L being the first number you turn up is
.
However, your system says that you ignore the first number and proceed. This means that
at this point you have a chance of being wrong and losing.
.
Next look at the possibility that you draw M then S then L. By the rules of your "system" you
discard the M, then draw the S, which being smaller than M, you also discard, and you end
up drawing L on your last turnover. You win. There is a chance of this happening.
.
Next possibility. You draw M then L and then S. By your system you draw M, discard it,
then draw L, and you quit without turning over the third number. You win!. There is a
chance of this outcome.
.
Next possibility. You draw S then M then L. By your system you draw S, discard it, then
draw the M. This being larger than S, you quit ... and you lose because L has not been
drawn.
.
Final possibility. You draw S then L then M. Again by your system, you draw S, discard it,
then draw L. The L being larger than S tells you to quit. You do, and you win! There is
a chance of this happening.
.
Adding it up, you see that you have 3 winning chances [MSL, MLS, SLM] each with a probability
of for a total winning probability of .
.
Your losing chances consist of the two chances that L is the first draw and the chance that
the draw sequence would be SML and you stop at M. This means that the total losing
possibility is
.
So your winning chance using the system is fifty-fifty, a significant improvement
over drawing the first number and quitting.
.
Hope this helps you to understand the problem a little better.