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Let's see if I understand this correctly. They each paid $9 for a total of $27. Of that amount
$2 went to the clerk and $25 went for the room. What missing dollar?
.
This is an old, old puzzle that never fails to confuse people. You subtract the amount that
is given to the clerk, not add it to the amount they paid ($27) in an attempt to make it
add up to $30.

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Just set f(x) equal to zero. When you do that the given equation becomes:
.

.
Now get rid of the denominators by multiplying all terms (on both sides) of this equation
by . When you do that multiplication you get:
.

.
Note that the right side of the equation remains zero because 0 times anything is still
zero. Note also that you can now cancel the terms in the denominator with the same term
in the numerator. The result is:
.

.
And the equation reduces to:
.

.
Multiply out the terms to get:
.
3x - 6 + 4x - 4 = 0
.
Combine the x terms and the equation becomes:
.
7x – 6 – 4 = 0
.
Combine the -6 with the -4 and the equation further becomes:
.
7x – 10 = 0
.
Eliminate the -10 from the left side by adding +10 to both sides. This makes the equation:
.
7x = 10
.
Finally, solve for x by dividing both sides of the equation by 7 which is the multiplier
of the x term. The division results in the answer:
.

.
Hope this helps you understand the problem.

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Your answer is correct for this one also. You're doing great!

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Your answer is correct! Keep it up!

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You are correct. When the factors of answer (c) are multiplied together the product is
the original trinomial that you were given to factor.
.
Good job!

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Let me start by saying that these two equations have no common solution. You can tell that
because when you graph them their graphs turn out to be parallel lines. In order for linear
equations such as these to have a common solution, their graphs must cross at some point.
.
You can get points on each of the graphs easily. For the first equation:
.
3x - y = 1
.
one point could be found by setting x equal to zero. When you do that, the equation
becomes - y = 1. Solve for y by multiplying both sides by -1 to get y = -1. This tells
you that (0, -1) is a point on the graph. When you plot this point, you will find that it
is on the y-axis. [by setting x = 0, the answer for y will be on the y-axis.]
.
Another point on the graph can be found by setting y = 0. When you do that the equation
reduces to 3x = 1. Solve for x by dividing both sides by 3 and you get x = 1/3. In this
case since x = 1/3 when y = 0, you know the point (1/3, 0) is on the graph. Note that when
you plot this point you will find that it is on the x-axis. [by setting y equal to zero,
the answer for x will be on the x-axis.]
.
Plot those two points (0, -1) and (1/3, 0) and extend a straight line that runs through
them and you have the graph of the first equation.
.
You can get two points for the second equation in the same way. In the second equation:
.
3x - y = 2
.
You can set x equal to zero and solve for y to find that y = -2. So the point (0, -2) is
on the graph of the second equation.
.
Then set y = 0 and solve for x to find that the point (2/3, 0) is also on the graph of
the second equation.
.
Plot these two points and extend a straight line through them to complete the graph of
this equation.
.
Your graphs should now look like this:
.

.
Notice that the "brown" graph is for the first equation because it crosses the y-axis
where y equals -1 and the "green" graph is for the second equation because it crosses
the y-axis at -2.
.
Notice that the lines are parallel, so they will never cross. Therefore, there is no common
solution for the two equations.
.
Hope this helps you to understand the process of finding a common solution (or lack of
a common solution) by using the method of graphing.

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Yes, you got it correct.
.
You can always check your answer by multiplying the two factors you got using the FOIL method.
.
FOIL stands for Firsts ... Outsides ... Insides ... Lasts and it applies to the arrangement
of the terms in your answer. Your answer was:
.
(b – 3a)(b + 2a)
.
The two "Firsts" in each set of parentheses are both b. Multiplying them together gives you
b*b which is b^2.
.
The two "Outsides" in this string are ... b from the first factor and +2a from the second
factor. Multiplying b times +2a results in +2ab.
.
The two "Insides" are -3a from the first factor and b from the second factor. Multiplying
these two gives you -3ab.
.
And finally, the two "Lasts" are -3a from the first set of parentheses and +2a from the
second set. Multiplying -3a times +2a results in -6a^2.
.
Now add your four answers from the FOIL process:
.
b^2 + 2ab - 3ab - 6a^2
.
The two middle terms (both containing ab) combine to give you -ab. Substitute this result
in place of +2ab - 3ab and your product becomes:
.
b^2 - ab - 6a^2
.
This is the same as the trinomial you were asked to factor. Therefore, you have factored
it correctly. Had your FOIL multiplication given you a different answer you might have
expected that your factors were wrong (or you made a mistake in the FOIL multiplication).
.
Hope this helps you to build confidence in what you are doing. Keep up the good work!

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Yes, your answer is correct!
In your factors:
a) the +3 times the -10 gives you the -30 you need and
.
b) the +3 plus the -10 gives you the -7 for the x term that you also need
.
Good job!

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Given:
.

.
Subtract 11 from both sides to get the constant on the right side of the equation. This results
in the equation becoming:
.

.
You want the multiplier (coefficient) of the term to be 1. To make that happen,
divide every term on both sides of the equation by the multiplier of the which is 2.
When you do that division the equation becomes:
.

.
Next, divide the multiplier of the x term by 2. That is divide the 5 by 2 to get .
Square that term to get . Now add to both sides of the equation.
You get:
.

.
The left side factors to:
.

.
Multiply the by to get which now has a common denominator
with the other term on the right side. Substitute this into the equation to get:
.

.
The right side combines to give:
.

.
Now take the square root of both sides to get:
.

.
Note that when you take the square root of the right side, the square root is preceded by
both a + and a - sign. This shows in the next step. Look carefully for the two signs
between the two terms on the right side.
.
Subtract from both sides and also transform the square root of 3/4 into the
square root of 3 divided by the square root of 4. When you do that you will get:
.

.
The square root of 4 is 2 and when this is substituted the equation becomes:
.

.
Hope this helps you to understand the process of completing the square to solve a quadratic
equation.

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Given f(-8) = 200 and f(30) = 580
.
Find an exponential formula that fits these given data.
.
The formula you will need to find is of the form
.
So what you need to do is to solve for A and b using the given data and appropriate
math processes.
.
Start with the given that y = 200 when x = -8. Plug these values into the formula to get:
.

.
Take the natural logarithm of both sides:
.

.
But the natural log of e is 1. Substitute that and you get:
.

.
Solve this for ln(A) by adding 8*b to both sides to get:
.

.
Next follow the identical process for the second data point in which y = 580 and x = 30.
Using this data in the formula and solving for ln(A) results in:
.

.
Now you can set the right sides of the two equations you have for ln(A) equal and solve
for b. You should be solving:
.

.
When you solve this you should find that b is somewhere around 0.028018703 or whatever
you choose to round it to.
.
Now that you know b you can return to either of the two equations for the data points and
solve for A. For example, return to:
.

.
Substitute 200 for y, -8 for x, and 0.028018703 for b to get:
.

.
Solve for A by dividing both sides by
.

.
But a quantity in the denominator that has a negative exponent can be brought into the
numerator with the same exponent only positive. This means that A becomes equal to:
.

.
and by a calculator . Substituting this into the equation
for A results in:
.

.
Now put this value of A and also the known value of b into the equation and you should
end up with something around:
.

.
as the equation you are looking for.
.
Hope you can track your way through this. You can check by letting x = 30 and solving
to see if y = 580. You should find that y computes to be 579.9999895 or so. Close
enough to assume this works. Cheers and good luck!

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There are several ways that you can find the value of x that represents the axis of symmetry.
.
One of these ways is to get the quadratic equation in the form that you need to apply the
quadratic formula. That form is:
.

.
Then recognize that the axis of symmetry is determined by that term of the of the answer to
the quadratic equation that is:
.

.
In this case, by comparing the quadratic expression of the problem:
.

.
with the standard quadratic form:
.

.
you can see that a = 1, b = -1, and c = -8. Substitute the appropriate values here into:
.

.
and you get:
.

.
So the answer to this problem is that the equation for the line of symmetry is:
.

.
Hope that this clarifies the significance of the first term in the quadratic formula:
.

.
It is the axis of symmetry about which the answers are equally spaced in both the + and
the - directions.
.

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First, let B represent the number of acrobats wearing blue. Also let P represent the number
of acrobats wearing pink.
.
Two times the number of acrobats wearing pink is, therefore, 2*P
.
1200 less that that is 2*P - 1200 and that is equal to the number of acrobats wearing
blue. So we can write that B = 2*P - 1200
.
So we can write the ratio of blue to pink acrobats as:
.

.
But you were told in the problem that . Substitute this into the ratio
equation to get:
.

.
An easy way to work on this proportion (a fraction on the opposite sides of the equal sign)
is to cross multiply ... that is multiply the numerator on the left side by the denominator
on the right side and set this product equal to the product you get by multiplying
the numerator on the right side by the denominator on the left side. For this problem
that translates to multiplying 4 by P and setting that equal to multiplying (2*P - 1200)
by 5. This gives you the equation:
.

.
Multiply out the right side and the equation becomes:
.

.
To get the P terms on the same side of the equation subtract 10P from both sides of the
equation. This results in:
.

.
Solve for P by dividing both sides of the equation by -6 which is the multiplier of P.
When you do this division, the result is:
.

.
You now know that there are 1000 acrobats in pink. Since the ratio of Blue to Pink is 4 to 5,
you can write:
.

.
Using the cross product process just as before results in:
.

.
Dividing both sides by 5 results in:
.

.
So there are 800 acrobats dressed in blue and 1000 acrobats dressed in pink.
.
Check ... twice the acrobats dressed in pink is 2000. Notice that the number dressed in
blue is 2000 - 800 = 1200 less than that number. That checks. And the ratio of 800 acrobats
in blue to 1000 in pink is and that reduces to so it also checks.
.
Hope this helps you to see how to work the problem.

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If you meant the problem to be:
.

.
Then your answer is correct!
.

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Your answer is correct. Good job!

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Go to the following website:
.
en.wikipedia.org/wiki/Algebra
.
and read the first sentence. It explains it ...

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Begin by writing the equations for the Cost (C) of each printer. The general form of these
equations is based on multiplying the cost per page times the number of pages and adding
to that the fixed cost that you pay regardless of the number of pages.
.
For the first printer, the equation for the Cost (C) and the number of pages (P) is:
.
C = (0.02*P) + 5
.
For the second printer, the equation for the Cost (C) is:
.
C = (0.015*P) + 7
.
The first printer is cheaper for a while. You can tell that because if you have 1 page
printed, the cost at the first printer is $5.02 and the cost at the second printer is
$7.015. However, as you increase the number of pages, the difference in the Cost between
the two printers gets smaller and smaller until at some number of pages the Cost of the
two printers is the same. After that number of pages, the Cost for printing more pages
will be less at Printer 2.
.
You can find the point at which the number of pages make the Cost equal at the two printers.
Do this by solving the two Cost equations simultaneously.
.
The two equations are:
.
C = (0.02*P) + 5
C = (0.015*P) + 7
.
We are trying to find the number of pages (P) that will make the Costs equal. Notice that
if C is the same in both equations then the right side of both equations must be equal
also. So we can say:
.
(0.02*P) + 5 = (0.015*P) + 7
.
To solve this requires collecting the terms that have P on one side of the equation
and the numbers on the other side. You can start by subtracting (0.015*P) from both sides
of the equation. This makes the equation become:
.
(0.02*P) - (0.015*P) + 5 = + 7
.
Next subtract 5 from both sides. This results in:
.
(0.02*P) - (0.015*P) = + 2
.
Then subtract the two terms on the left side to get:
.
0.005*P = 2
.
Finally, solve for P by dividing both sides by 0.005, the multiplier of P. The solution
for P is:
.
P = 2/0.005 = 400
.
So if you have 400 pages to print, the Cost should be equal at the two printers. Check this
out.
.
The Cost at the first printer would be $5.00 plus 2 cents a page for the 400 pages
which is 800 cents or $8.00. So the total cost is $5 + $8 = $13.
.
The Cost at the second printer would be $7.00 plus 1.5 cents a page for 400 pages
which is 600 cents or $6.00. That makes the the total Cost $7 + $6 = $13.
.
Since the Cost is $13 at each printer for 400 pages, you can draw the following conclusion:
.
For less than 400 pages, it will be cheaper to use the first printer. At 400 pages it
doesn't matter which printer you use, the Cost will be the same. For any number of pages
greater than 400, you will save money by using the second printer.
.
Answer C is the correct answer.

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Given: an unknown four-digit number.
.
Call the first digit x. Do this because the second digit is given in terms of the first.
And the third digit is in terms of the second which is based on the first. And the third
is based on the second which depends on the first. And the fourth is based on the third,
which is based on the second, which depends on the first. Therefore, every one of the
four digits is based on the first digit in some way. So if you can find the unknown
first digit, you should be able to find all the other digits.
.
This first digit is 4 more than the second digit. This means that if you took 4 away from
the first digit the answer would be the second digit. Therefore, you can write the second
digit as x - 4
.
The second digit is 3 less than the third. This means that if you add 3 to the second digit,
the answer would be the third digit. Since the second digit is x - 4, adding 3 to it results
in (x - 4) + 3 = x - 4 + 3 = x - 1. Therefore, the third digit is represented as x - 1.
.
Finally, the fourth digit is 1 less than the third. That means that if we subtract
1 from the third digit the result is the fourth digit. Since the third digit is x - 1,
taking 1 away from it gives (x - 1) - 1 = x - 1 - 1 = x - 2.
.
Let's summarize to this point:
.
First digit = x
Second digit = x - 4
Third digit = x - 1
Fourth digit = x - 2
.
The problem says that the sum of these digits is 21. Write this in equation form:
.
x + (x - 4) + (x - 1) + (x - 2) = 21
.
Since each set of parentheses is preceded by a plus sign, you can just remove the
parentheses without any changes to the terms inside each set. (I just put the parentheses
in the equation to separate each of the four numbers.) So remove the parentheses to get:
.
x + x - 4 + x - 1 + x - 2 = 21
.
Adding all the x terms together reduces the equation to:
.
4x - 4 - 1 - 2 = 21
.
Now add the three negative numbers together to get:
.
4x - 7 = 21
.
Get rid of the -7 on the left side by adding +7 on the left. But if you do that you also
have to add +7 to the right side to keep the equation in balance. Adding +7 to both
sides makes the equation become:
.
4x = 28
.
Finally solve for x by dividing both sides of the equation by 4 which is the multiplier
of the x. When you divide both sides of the equation by 4 it becomes:
.
x = 7
.
Since x is the first digit of the 4 digits, you can now solve for the other three
digits.
.
You know that the second digit is represented by x - 4. Substituting 7 for x, the second
digit becomes 7 - 4 and the answer to that is 3. So the second digit is 3.
.
The third digit is represented by x - 1. Substituting 7 for x results in 7 - 1 which
simplifies to 6. So the third digit is 6.
.
Finally the fourth digit is represented by x - 2. Substituting 7 for x gives you 7 - 2 which
results in the fourth digit being 5.
.
The number is 7365.
.
Let's check it against the problem statement.
.
Is the first digit 4 more than the second digit? Yes, 7 is 4 more than 3.
.
Is the second digit 3 less than the third digit? Yes, 3 is 3 less than 6.
.
Is the fourth digit 1 less than the third digit? Yes, 5 is 1 less than 6.
.
Finally, is the sum of the 4 digits equal to 21? Yes, 7 + 3 + 6 + 5 = 21
.
Everything checks, so the answer of 7365 is correct.
.
Hope this helps you to understand the problem a little better.

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You don't have to get involved with joules or units other than those mentioned in the problem.
.
Look at the problem this way. In one hour the bulb consumes 60 watts times 1 hour which
is the same as 60 watt*hours. To convert this to kw*hours, just divide by 1000. When you
do this division you get:
.
60 w*hr/1000 w/kw = 0.060 kw*hr
.
Units check:
.

.
So for each hour it runs, the bulb uses up 0.060 kw*hr of energy.
.
How many hours does the bulb run per year? At 8 hours per day for 365 days in the year, the
bulb runs 8 times 365 hours which is 2920 hours.
.
Units check:
.

.
Then in a year the total consumption is 0.060 kw*hr for each hour times the 2920 hours in
a year for an annual amount of:
.
kw*hr per year.
.
Units check:
.

.
And at a cost of $0.0683 for each kw*hr the annual cost to run the bulb will be:
.
$0.0683 * 175.2 = $11.96616 which rounds to $11.97.
.
Units check - using D to represent dollars):
.

.
Everything checks. Hope this helps you to see your way through this problem.

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Presuming that you don't mean that only the square root of 24 is divided by 2, your problem
is to find:
.

.
This has two answers ... one answer involves the plus sign and the other answer involves
the minus sign. In other words, the answers to your question are:
.
and
.
You can find using a calculator. You should find that
.
Substituting that number for in the two answers results in the two answers
becoming:
.
and
.
In the first answer the two terms in the numerator add to give 11.898979486 so this answer
becomes:
.
and dividing the 2 into the numerator of 11.898979486 results
in that answer becoming:
.

.
In the second answer the two terms of the numerator are subtracted. This subtraction
(7 - 4.898979486) results in 2.101020514. Substituting this value for the numerator
of the second answer, results in the second answer becoming:
.

.
In summary, your original problem has two answers:
.
and
.
Hope this work helps you to understand how to evaluate expressions of the form that you
have in this problem.

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Given:
.

.
If you want, you can solve this without using the quadratic formula. Notice that the left
side of this equation simplifies to:
.

.
So you can substitute for the left side of the equation you were given. This
makes the given equation become:
.

.
Then you can find by dividing both sides by 4 which is the multiplier of the
term. This division of both sides by 4 reduces the equation to:
.

.
and
.
Substituting for results in the equation becoming:
.

.
and taking the square root of both sides gives you the two answers:
.
and
.
And since the square root of 10.5 is (from a calculator) 3.240370349, the two answers to
this problem are:
.
and
.
But if you have to do the problem using the quadratic formula, you need to get the
original problem into the standard quadratic form of:
.

.
You still combine the two terms on the left side of the original equation to get:
.

.
just as we did above. Next you need to get zero on the right side as the standard
quadratic form calls for. Do this by subtracting 42 from both sides of the equation.
That subtraction results in:
.

.
This is in the standard quadratic form in which a = 4, b = 0 (because there is no term
that contains just x), and c = -42.
.
The quadratic formula says that the answer to the problem is determined by the equation:
.

.
Since b = 0 in this problem, the equation reduces to:
.

.
Next substitute 4 for a and -42 for c to get:
.

.
The term in the radical [that is -4*4*(-42)] multiplies out to be 672 so the problem becomes:
.

.
The denominator of (2*4) becomes 8 which further simplifies the problem to:
.

.
Since the square root of 672 is (by a calculator) 25.92296279 the answers to this
problem become:
.
and
.
When you do the division, the two answers are:
.
and
.
And these are the same answers that we found earlier without using the quadratic formula.
.
Hope this discussion helps you to work your way through the problem.

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Given:
.
(6 + 8i) - (-7 + i)
.
The first set of parentheses is preceded by a plus sign (you can presume that because
there is no sign actually shown). Parentheses that are preceded by a + sign (or by no sign)
can be removed without any change to the terms they contain. This changes the given
problem to:
.
6 + 8i - (-7 + i)
.
Notice that the remaining set of parentheses is preceded by a minus sign. The rule is that
you can remove these parentheses if you change the signs of the terms inside the parentheses.
Following this rule leads to the given expression now becoming:
.
6 + 8i + 7 - i
.
Now you deal with the "regular" integer terms separately from the terms containing "i".
In other words you can combine the +6 and +7 to get +13. Then you add the 8i and the -i
to get +7i.
.
The answer to your problem is the combination of these two terms and it is:
.
13 + 7i.
.
Hope this helps you to work your way through this problem and helps you understand
how the process works.
.

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Given:
.
-2y = 3x - 15
.
You can easily find the slope and the y-axis intercept if you can get the given equation
in the form:
.
y = mx + b
.
In this form the slope is "m" and the y-axis intercept is "b"
.
To get the given equation into slope intercept form, you need to solve for +y. In this case
the left side will become +y if you divide all terms on both sides of the given equation
by -2. The process is as follows:
.
-2y = 3x - 15 <--- given equation
.
(-2y/-2) = (3x/-2) - (15/-2) <--- shows division of all terms by -2
.
y = -(3/2)x - (-15/2) <--- notice that - (-15/2) = + 15/2
.
y = -(3/2)x + (15/2) <--- given equation in slope intercept form
.
By comparing this equation with the slope intercept form, you can see that "m" (the
multiplier of the "x") is -3/2 and "b" the y-axis intercept is +15/2. That's the answer
to the problem. As an aid to seeing that this is correct, here's a graph of the original
equation in the problem:
.

.
Notice that the slope is negative and the value of y where the graph crosses the y-axis is
about +7.5 or +15/2.
.
Hope this helps you to understand the problem a little more.

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By convention:
.

.

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Hmmmm. I wonder if you are doing this as a math exercise or as an actual project. As a
math exercise you can make certain assumptions that you may not want to make if you are
performing a "real world" measurement.
.
Let's do it as a math exercise first. The circumference of an NBA basketball is (measured
in the grooves on the ball) 29.5 inches. Use the formula for circumference to find the radius
of an NBA basketball:
.

.
Substitute 29.5 inches for the circumference of the ball and you get:
.

.
Solve for R by dividing both sides by to get:
.

.
Using a calculator to do the math on the left side results in:
.
inches
.
Now you can find the volume of the basketball by using the formula for the volume of
a sphere:
.

.
Substitute 4.695070821 inches for R and the equation becomes:
.

.
Calculator time ... Using a calculator to perform the math on the right side of this equation
you should find that:
.
cubic inches
.
Now assume that 12 oz of root beer is the same volume as 12 oz of water. At 39.164 degrees
Fahrenheit water weighs 0.036313 lbs per cubic inch and since a lb is 16 oz, we can multiply
this figure by 16 to find out that the density of water is:
.
ounces per cubic inch. Therefore a 12 oz can contains
a volume of cubic inches.
.
[Just as a sanity check, the approximate dimensions of a 12 oz can are about 1.25 inches
in radius and 4.5 inches tall to the lid. This leads to a volume of cubic
inches ...
pretty close to what we found.]
.
So if a 12 oz can of root beer contains 20.65376036 cubic inches of root beer, we can
divide this into the volume of the basketball, and find out how many cans the basketball
can hold. Recall that the volume of the basketball was 433.5259036 cubic inches. Divide
that by 20.65376036 cubic inches per can of root beer, and the answer for the number
of cans of root beer that can be poured into the basketball is:
.
or about 21 cans of root beer in the
basketball.
.
There are a lot of chances for error in this problem. Recall that we used the outside
dimensions of the basketball to calculate the interior volume. This ignores the fact that
the interior of the basketball will be somewhat smaller because of the thickness of the skin
that forms the ball. Also the NBA basketball is allowed to range in circumference
from 29.5 to 29.75 inches. So its interior volume can vary somewhat from its size and
the thickness of the material that comprises. Add to that that the interior volume of
the basketball is less when the ball is deflated so you can pour root beer into it.
.
Another source of error can come from the fact that we equated ounces of root beer to
ounces of water. This is not likely to be true. A can of root beer will likely sink in water,
indicating that it is denser than water. [However, a can of diet root beer may very well
float because it is likely to be less dense than water.]
.
And finally, the density of water changes slightly with temperature. The temperature
of the water we used makes water its most dense. Warmer than that temperature would
decrease the density.
.
All these variables would make this an interesting science experiment. However, the answer
we came up with is probably pretty close.
.
Check the math above. It's sort of late to be working math problems, and there's a reasonable
chance that an error may have crept in somewhere.
.
Hope this helps you to see your way through the problem.
.

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For lines to be parallel, they must have the same slope.
.
You are given the equation:
.
-5y=10x+32
.
You can find the slope of the graph of this equation by "rearranging" the given equation
into the slope intercept form of y = mx + b. [In this form m (the multiplier of x] is the
slope and b is the value on the y-axis where the graph crosses (or intercepts).
.
So we need to solve the given equation for y. If we do that, the resulting equation
will be in the slope intercept form. Since the given equation has -5y, we can find y
by dividing all terms on both sides by -5, the multiplier of y. Dividing by -5 results in:
.
y = (10/-5)x + (32/-5)
.
The division causes the coefficient [or multiplier] of x to become -2 and the resulting
equation is:
.
y = -2x - (32/5)
.
Since this is in slope intercept form, we now know that the graph of the given equation
crosses the y-axis at -32/5 and the graph has a slope of -2 (it goes vertically down 2
units every time it moves horizontally to the right 1 unit).
.
Since the line we are looking for is now known to have a slope of -2 (it must have the
same slope as the given line to be parallel), we can now write part of the slope intercept
form of the equation of this new line. The new line will be of the form:
.
y = -2x + b
.
The problem tells you that this line must go through the point (-1,1) which means that
the equation must be true when x = -1 and y = +1.
.
So we can substitute these values for x and y in the equation of the new line. Therefore,
substitute these values into:
.
y = -2x + b
.
and you get:
.
1 = -2(-1) + b
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Multiply out the first term on the right side and you get:
.
1 = 2 + b
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Solve for b by subtracting 2 from both sides of the equation to get:
.
1 - 2 = 2 - 2 + b
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which simplifies to:
.
-1 = + b
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We now know that b = -1 so our equation for the new line becomes:
.
y = -2x + b
.
and by substituting -1 for b it is:
.
y = -2x - 1
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That's the answer. And if you graphed the original equation and this new equation,
the graph would look like this:
.

.
in which the green line (which crosses the y-axis at -1) represents the parallel line that
we found, and the other line is the graph of the equation you were originally given.
.
Hope this helps you to understand the process of making a parallel line. Remember
that the slopes of the two lines must be the same.
.

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Given:
.
x - y = 4
.
Subtract x from both sides to get rid of the x on the left side of the equation. Subtracting
x from both sides results in:
.
- y = -x + 4
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But you need to solve for +y to get the equation into the slope intercept form. So change
the minus y to plus y by multiplying all terms on both sides of the equation by -1. When
you do that the equation becomes:
.
y = +x - 4
.
And that's the answer. It is in the form of y = mx + b in which m = +1 and b = -4.
The graph of this equation has a slope of +1 and it crosses the y-axis at -4. The graph looks
like this:
.

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

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You can do this problem without using logarithms if you wish. The process for doing it
involves:
.
First: use the power rule of exponents to raise to the x power. By this rule the
exponent "x" is applied to both the numerator and the denominator so the given equation becomes:
.

.
But 1 raised to the "x" power is always 1 no matter what the value of "x" is. So the equation
simplifies to:
.

.
Notice that the numerators are equal ... both of them are 1. So for this equation to be
true, the denominators must also be equal. Therefore you can say that:
.

.
Six squared is 36 and if you multiply that by 6 you get 216. So you can say that
6 cubed = 216 so "x" must be 3 to indicate that 6 is cubed.
.
If you have to do the problem by using logarithms, then start with the given equation:
.

.
and take the logarithm of both sides:
.

.
Now apply two rules of logarithms. The first rule is that in a logarithm of a quantity
that is raised to an exponent, the exponent can be brought out to become the multiplier
of the logarithm. Applying that rule to the left side of the equation, makes the equation
become:
.

.
The second rule of logarithms to apply is the one that says the log of a fraction is
equal to the log of the numerator minus the log of the denominator. Applying that rule
to both sides in this problem gives:
.

.
Now use the fact that log(1) = 0 and substitute 0 for log(1) on both sides to get:
.

.
and this simplifies to:
.

.
If you multiply both sides of this equation by -1 you get rid of the minus signs and the
equation becomes:
.

.
Divide both sides of this equation by to solve for x. This division
leads to:
.

.
Now use your calculator to find that for base 10 logs and
. Substitute these two values to get:
.

.
And when you do this division on a calculator you get .
.
Hope this helps you to understand a little more about both logarithms and also about handling
exponents.
.

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Is:
.

.
a perfect square? The answer is, "No, it is not."
.
But how do you easily get that answer. The method to use is called "Completing the Square."
.
How do you use that method? First, make sure that the coefficient of the term
is 1. [In this case it is, otherwise you would have to factor out from all the terms in
the trinomial the coefficient (multiplier) of the term.]
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Next, forget about the +48 for the time being. Take half of the multiplier of the x term. That
multiplier is -24 and half of it is -12. Then square the -12 to get + 144.
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Now we can say that is the perfect square trinomial. [Its equivalent
form is .] So for the given trinomial to be a perfect square, the +48 would
have to be changed to +144.
.
Another way to do this (maybe even easier) is to use the discriminant of the quadratic
formula. The quadratic formula applies to all equations of the general form:
.

.
The discriminant of this equation is given by . If the discriminant
is equal to zero, the trinomial is a perfect square (the equation has two real and
equal roots).
.
By comparing your problem with the general form of the quadratic equation you can see
that a = 1, b = -24, and c = +48. Substituting these values into the discriminant,
you get:
.

.
This simplifies to:
.

.
Since the discriminant does not equal zero, the given trinomial is not a perfect square.
.
However, suppose that the given trinomial had been . In this case
a = 1, b = -24, and c = 144.
.
Substitute these values into the discriminant and you get:
.

.
And since the discriminant now equals zero, this trinomial is a perfect square. [As we
noted earlier the trinomial is equivalent to .]
.
Hope these couple of ways help you to understand your problem better.

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Find the equation for the surface area of a cube with an edge of x centimetres long.
.
Each face of the cube is a square that has a length of x cm for a side. The area of a
square is found by squaring the length of a side. So the area of a face of this given
cube is . Think about the shape of the cube. (A die is an example.) It has
a total of 6 faces ... 4 going around it plus a top face and a bottom face. Since there are
6 faces and each face has an area of , the total surface area (call it S) is given
by:
.

.
and since the given dimensions for the side of a face is in centimetres, the units of the
answer is square centimetres
.
To find the volume of a cylinder, you multiply the area of its circular base times the
height of the cylinder. The area (A) of a circle is given by the equation:
.

.
where R is the radius of the circle. In this problem you are told that the radius is
given by x, so substitute x metres for R and the equation becomes:
.

.
Then, to find the volume (V) you multiply this area by the height (H) and you get:
.

.
but you are given that the height of the cylinder is 5 m. Substituting this value for
H results in the volume equation becoming:
.

.
and since the radius and height are in metres, the units of the answer is cubic metres.
.
Hope you can track the development of each of these equations and that it helps you to
understand the problems a little better.

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First Problem. Given:
.

.
To get this into slope intercept form, you need to solve for y. This means that you need
to get the y-term by itself on one side of the equation. So you need to start by getting
rid of the x term on the left side of the given equation. You can do this by subtracting
x from the left side, but if you do then you must also subtract x from the right side too
to keep the equation in balance. So subtract x from the left AND right sides to get:
.

.
But you are trying to solve for y and you have 2y on the left side. So divide the left
side by the multiplier of y (which is 2). However, if you divide the left side by 2, then
you must also divide both terms on the right side by 2. Dividing both sides by 2 results in:
.

.
This is the slope intercept form of the given equation. You can tell from this form that
the slope is and the graph crosses the y-axis at a value of +3.
.
You seem to have had the correct idea, but made an error in the sign of the slope.
The minus sign means that the graph goes down as you move to the right on the graph, and
your answer of + would have the graph moving upward as you move to the right on
the graph. The graph below shows the correct graph:
.

.
You can quickly check the graph by returning to the original equation for this first
problem:
.

.
Set x equal to zero and this equation becomes . Then divide both sides by 2 and the
equation reduces to . This tells you that the (x,y) point (0,3) should be
on the graph. Next set y equal to zero and the equation reduces to . This
tells you that the (x,y) point (6,0) should also be on the graph. I think you'll find
that (0,3) and (6,0) are both on the graph ... which helps to verify that the graph is
correct.
.
Second Problem. Given
.

.
The same general process as we used in the first problem applies here. Solve for +y
by subtracting 2x from both sides of the given equation. This subtraction leads to:
.

.
This is in slope-intercept form. The slope is -2 and the intercept is +9. The graph is:
.

.
You can again check the graph by returning to the original equation for this second
problem. This equation was:
.

.
Set x equal to zero in the equation and the equation becomes . So one (x,y) point on
the graph should be (0,9). Then set y equal to 0 and the equation becomes .
When you then divide both sides by 2 you get . So the (x,y) point (4.5, 0)
should be on the graph too.
.
It appears as if you have the correct idea, but just made the same sign error in getting
the equations for the two problems into the slope-intercept form. Notice that since both
of these problems resulted in a negative slope, the two graphs go down as you move to
the right along the graph.
.
Hope this helps you to understand the process a little better and that the discussion
helps you too.

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Given:
.

.
Also given that the voltage (V) is 120 volts, and the Power (P) is 1500 watts.
.
Substitute these values into the equation to get:
.

.
Square both sides of this equation ...
.

.
Squaring the 120 results in 14400. Substituting this results in:
.

.
To solve for R divide both sides of this equation by the multiplier of R which is 1500:
.

.
The division on the left side results in:
.

.
The answer is that the resistance R is equal to 9.6 ohms.
.
Hope this helps you to understand the math manipulations that can be used to solve this
problem.