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x - y = 12;

(15,3): 15-3=12 ->->-> It is a solution!

(9,6): 9-6 = 3 ->->-> It is NOT a solution!

(18,6): 18-6=12 ->->-> It is a solution!

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The monk goes up to the middle step. Then he goes up 8 more steps and down 12 steps, which places him 4 steps below the middle step. From this step, he goes UP 27 steps to the top. This means that there are 27-4=23 steps from the middle step to the top. There must be a total of 2*23 = 46 steps.

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The first step is to find the LCD, which is .

Multiply the first numerator and denominator by "ad", and the second numerator and denominator by "3c". This brings each denominator up to the common denominator.



The numerator could be factored, but the fraction does not reduce so factoring is not necessary.

If you have trouble with this, please see my own website by clicking on my tutor name "rapaljer" anywhere in algebra.com, and look for "MATH IN LIVING COLOR", "Basic Algebra", "Chapter 3", "Least Common Denominator", and "Adding and Subtracting Fractions."

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There is NO REAL SOLUTION, since there is no real number that you can raise to an even power and get a negative result. If imaginary (complex) solutions are allowed, then there will be solution(s), but NOT for real numbers.

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The number of subsets is always 2 raised to the power of the number of elements in the set. In this case, the answer is .

The number of proper subsets is always 1 less than the number of subsets, since a proper subset is any subset except the set itself! In this case, the answer is .

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What you have here is 7 equations and 7 unknowns, unless someone sees a faster way to do this!!

Let the 7 variables be x1, x2, x3, x4, x5, x6, and x7 respectively.

Now, here are the equations:
x1+ x2+ x3+ x4+ x5+ x6+x7=200
x1+x2=57
x2+x3=83
x3+x4=71
x4+x5=43
x5+x6=66
x6+x7=43

If you have a graphing calculator, like a TI85, 86, 83+, or 84, you may have a program that will solve this system called [SIMLT] or perhaps [POLYSMLT].

If you don't have a calculator, then try getting everything in terms of x1. Do this by starting with
x2 =57-x1

Then
x3 = 83-x2
x3=83-(57-x1)
x3=26+x1

Next,
x4= 71-x3
x4=71-(26+x1)
x4=45-x1

x5=43-x4
x5=43-(45-x1)
x5=-2+x1

x6=66-x5
x6=66-(-2+x1)
x6=68-x1

x7=43-x6
x7=43-(68-x1)
x7=-25+x1


Now,
x1+ x2+ x3+ x4+ x5+ x6+x7=200
x1+(57-x1)+(26+x1)+(45-x1)+(-2+x1)+(68-x1)+(-25+x1)=200
x1+169=200
x1=31
x2=26
x3=57
x4=14
x5=29
x6=37
x7=6

That should do it!!!

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I figure that the designer had 600 pieces. Check it out, by figuring that he/she had a square of 24 by 24, with 24 characters left over. That equals 600. Now, adding one unit to each side of the square, you have a 25 by 25 square, which is 625 square units. That leaves an extra 25 spaces left over.

The final answer is 600. As my attempt to explain how I arrived at this, consider the following well-known illustration that consists of a sequence of dots.

Begin with 1^2, and draw 1 dot.

Next, take 2^2, and draw 4 dots in a 2 by 2 square. You can do this by adding 3 dots to the previous dot.

Next, take 3^2, and draw 9 dots in a 3 by 3 square. You can do this by adding 5 dots to the previous square.

Next, take 4^2, and draw 16 dots in a 4 by 4 square. You can do this by adding 7 dots to the previous square.

Continue the pattern, and see if you can figure out the generalization. It turns out that if you had n^2 dots in the square, you must add 2n+1 dots to get the next square which is (n+1)^2 dots.

This 2n+1 dots represents the sum of the number of dots the designer had left over in the smaller rectangle PLUS the number of dots that were open in the larger rectangle. In other words,





Number of characters in the orginal square = .
Total number of characters = characters.

I'm sorry this one was hard to explain!!

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Divide both sides by -1:




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You must rationalize the denominator by multiplying numerator and denominator by .



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You have to state or describe the problem.

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Write with the highest powers of x first:
-x+8

The degree is the highest power of x, which in this case is 1.

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Since the order of the multiplication is reversed, this is the Commutative Property of Multiplication.

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Your question was probably, "Name the property used in the following equation." The answer is the Commutative Property of Addition, since the order of the addition within the parentheses is changed.

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X*X=4X+32


Because this has an in the problem, it is a QUADRATIC EQUATION, and you must first set it equal to zero:


Next, factor it:



Now, set each factor equal to zero:
or
or

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center:(5,-2) and radius:16



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First, this is a QUADRATIC EQUATION, so you must make sure the equation is set equal to zero, then you must factor it:


or
or

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First, this is a QUADRATIC EQUATION, so you must make sure the equation is set equal to zero, then you must factor it:


or
or

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We do not have your book! You will have to state the problem!!

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In this problem, the key word is "choose". When you "choose", usually the order is NOT important. Therefore, when the word "choose" is used, it almost ALWAYS means that it is a COMBINATION. (See the solution that I posted a few minutes ago explaining the difference between a COMBINATION and a PERMUTATION.

This will be a combination of 10 fruits, choosing 3 at a time, or a C(10,3). Again, most calculators will do this for you, but in case you need to do it without a calculator, you start out as with a permutation with 3 spaces:

Remember that with a permutation it will be:
P(10,3)= ___*___*___
P(10,3)= 10*9*8=720

Now, a combination is similar. Make 3 spaces.
C(10,3) = ___*___*___

What is different with a combination, is that you must ALSO divide by 3*2*1. It looks like this:
C(10,3) = **=

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An equation of a line passing through (-8,-4) with undefined slope must be a VERTICAL LINE. A vertical line can always be written in the form x= a number. In this case the "number" is -8, so x=-8 is the equation of the line.

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This is the SUM of x^2 and y^2 with UNLIKE coefficients, so it is an ELLIPSE!!

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Combinations and permutations really give students a hard time. First of all, with combinations, the ORDER does NOT matter. With permutations, the ORDER DOES matter!! In this case, what you are trying to count is the number of ways to arrange a batting ORDER!! Obviously, the ORDER matters, so this is a permutation.

You must choose a batting order choosing 9 people out of 13 players. This is a permutation of 13, taking 9 at a time. You could write this as
P(13,9). Most calculators will do this for you if you know how to enter it in the calculator. If you don't know how to do this, you can always calculate it yourself, by making 9 blank spaces in a product as follows:
P(13,9) = ___*___*___*___*___*___*___*___*___

Then start with 13, and count down, filling all the 9 spaces.
P(13,9) = 13*12*11*10*9*8*7*6*5=259,459,200.

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If the numbers on the spinners cannot be repeated, then each time you set a number on a spinner, then you can't use that number again. Therefore, there will be 8 possibilities on the first spinner, then (having used up one possibility) there will be 7 on the second spinner, 6 on the third spinner, and only 5 on the third.

The number of possibilities will be
8*7*6*5 = 1680 possible combinations.

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Since there are 8 different settings on each spinner, and there are 4 different spinners, there will be 8*8*8*8 = 4096 combinations.

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What kind of simultaneous equations are you solving? Are they linear? That is, just x and y equations, where the variables are raised to the first power? Or, are they what we call "non-linear", which includes variables raised to powers other than the first power? Why don't you check out my website by clicking on my tutor name "rapaljer" anywhere in algebra.com, and look for "MATH IN LIVING COLOR". Now, if you are looking for LINEAR equations, go to "Intermediate Algebra" and look in Chapter 5 for the "Section 5.04 Systems of Equations". If you need NON-LINEAR equations, go to "College Algebra" and look in Chapter 3 for "Section 3.10 Non Linear Systems".

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P.S. Thanks for using my website!!

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As Checkley correctly pointed out, these two equations represent the same straight line. However, if the solution is the SAME LINE, then the entire line is the solution to the problem. There are infinitely many solutions, since EVERY point on the line is a solution of both equations!!

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When I solved your problem, I did NOT get the same answer as the one posted by Checkley75. Here is what I got:


Multiply the first times everything in the second parentheses:


Next, multiply 2ab times everything in the second parentheses:


Last, multiply times everything in the second parentheses:


Now, combine like terms:

==>
=======>
__________________________________________


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The method you are asked to use for this problem is the method of substitution. You should begin by noticing that in this equation, if you let , then

Now, make these substitutions and you get:



This factors into

or

Don't stop here! You must still solve for the original variable which is x. Substitute these values of u back into the formula
or
or

Now, are you supposed to give REAL solutions, or are you supposed to also include imaginary (complex) solutions? IF COMPLEX solutions are allowed, then
or
or
, or

If you are finding ONLY REAL solutions, then the answer is only x=4 or x=-4.

Note: You don't really need the 0 in the answers above. It's just that in the algebra.com format, you can't write this + or - notation without using the 0! Sorry about that.

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f(x) = –3x – 2 is a straight line graph, with y intercept of -2, and slope of -3. It should look like this:



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