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# Recent problems solved by 'bucky'

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 Linear-equations/105743: Solve the following system of equations 2x+y=1 y=1-2x1 solutions Answer 76911 by bucky(2189)   on 2007-10-22 23:28:22 (Show Source): You can put this solution on YOUR website!This is a "trick" problem to see how observant you are ... You are given the two equations: . 2x+y=1 y=1-2x . Let's rearrange the bottom equation by adding 2x to both sides. When you do that the bottom equation becomes 2x+y = 1. But that is identical to the top equation. Therefore, the bottom equation and the top equation are identical. Every solution for the top equation is a solution for the bottom equation also. They are not independent equations and therefore they do not have a single, unique solution. . Hope this helps you to see your way through the problem and why the answer is what it is ... .
 Equations/105738: List all the prime numbers between 55-75. I am confused about Prime Numbers, exactly what are they in plain ole english could someone please explain. Thanks, Zack1 solutions Answer 76907 by bucky(2189)   on 2007-10-22 23:13:06 (Show Source): You can put this solution on YOUR website!A prime number is a number that can only be divided by itself or 1 to get an integer as an answer. For example is 3 a prime number? Yes it is because it can only be divided by 3 and 1 to get an integer as an answer. How about 6 ... is it a prime number? No it is not because it can be divided by 6, by 1, by 3, and by 2 and each of those divisions results in an integer as an answer. . You are to find all the prime numbers between 55 and 75. Note that both 55 and 75 are not prime numbers because both can be divided by 5. . 56 is not prime ... it can be divided by 2. From this you can deduce that any even number (except 2) is not a prime because it can be divided by 2. Therefore, only odd numbers can be a prime and every odd number is not a prime. And any number that ends in 5 is not prime because it can be divided by 5 and produce a quotient that is an integer. (2 is an exception to this rule. It is prime because even though it is even, it can only be divided by itself and 1 ... so it fits the definition of prime.) . The prime numbers between 55 and 75 are: 59, 61, 67, 71, and 73. Any other numbers in this range are divisible by something other than themselves and 1. Any of the other numbers in this range have at least a 2 or a 3 or a 5 as a divisor. . Hope this clarifies the term "prime" for you and helps you to see your way through the problem. . .
 Age_Word_Problems/105730: The equation V = -2000t + 21,000 describes the value in dollars of a certain model of car after it is t years old. If a car is worth \$11,000, substitute 11,000 into the equation to find the age of the car. Please explain step-by-step.1 solutions Answer 76901 by bucky(2189)   on 2007-10-22 22:45:58 (Show Source): You can put this solution on YOUR website!Given that the value of a car (V) that is t years old is given by the equation: . V = -2000t + 21,000 . The problem tells you that the value of a car is \$11,000 and asks you how many years old the car is ... where t is the age of the car. Substitute 11000 for V in the equation to get: . 11000 = -2000t + 21000 . Get rid of the 21000 on the right side by subtracting 21000 from both sides to change the equation to: . -10000 = -2000t . Now solve for t by dividing both sides of this equation by -2000 and you have: . -10000/-2000 = t . Doing the division on the left side results in: . 5 = t . The car is 5 years old. . Hope this helps you to understand this problem and how to find the answer. .
 Functions/105672: I cant solve this we are supposed to find the domain of this function (this is Algebra 1) y = 1 _____ x + 41 solutions Answer 76891 by bucky(2189)   on 2007-10-22 21:22:19 (Show Source): You can put this solution on YOUR website!Given: . . You are asked to find the domain of this function. The domain is all the values that x can take. . In this problem x can take any real value except x cannot equal -4. Why can't it be -4? . The answer is that if x is -4, then the denominator of the given fraction becomes zero ... -4 + 4 = 0. And since the numerator (which is 1) is divided by the denominator, this means that 1 would be divided by zero if x is equal to -4. But the rules of algebra tell you that division by zero is not allowed. . So the answer to this problem is that the domain for x is all real numbers from minus infinity to plus infinity except x cannot equal -4. . Hope this helps you to understand the problem. .
 Travel_Word_Problems/105693: If a car is traveling 55 miles per hour, how long will it take to go 12 miles? Is there a formula for this type of problem? Thank you!1 solutions Answer 76889 by bucky(2189)   on 2007-10-22 20:58:56 (Show Source): You can put this solution on YOUR website!The formula for this type of problem is: . Distance = Speed * Time . You might be able to understand this by thinking, "If I go at 60 miles per hour for 2 hours, how far do I travel?" The answer is: . Distance = 60 miles per hour* 2 hours = 120 miles . In the problem you are told that the distance to be covered is 12 miles. The speed is 55 miles per hour. Put these into the equation and you get: . 12 miles = 55 miles per hour * Time . You can then solve for the Time by dividing both sides of this equation by 55 to get: . 12/55 = Time . So it takes 12/55 hours to go 12 miles. The fraction 12/55 converts to a decimal by dividing 55 into 12 to get 0.218181818 of an hour. Since an hour has 60 minutes, you can convert the answer to minutes by multiplying 60 by 0.218181818 to get about 13.09 minutes. . So at 55 miles per hour you go 12 miles in a little over 13 minutes. . Hope this helps you to understand the problem and how to use the formula to solve it. .
 Finance/105594: Dolls cost \$140 per carton and trucks cost \$430 per carton. If an order comes in for a total of 100 cartons for \$28,500, what was the number of cartons of dolls? (Hint, Let T= trucks)1 solutions Answer 76854 by bucky(2189)   on 2007-10-22 12:43:20 (Show Source): You can put this solution on YOUR website!Let D equal 1 carton of dolls and T equal 1 carton of trucks. The problem says that the combined numbers of cartons is 100. Therefore, if we add D and T the total number of cartons is 100. In equation form this is: . D + T = 100 . Since the dolls cost \$140 per carton, if we multiply the number of cartons of dolls by \$140, we have the total amount of money spent on dolls. Therefore, \$140N is the amount spent on dolls. . Similarly, trucks are \$430 per carton. Therefore, the total amount spent on trucks is \$430 times the number of cartons of trucks that were purchased. . When you add these two amounts together the total is \$28,500. In equation form this is: . 140D + 430T = 28,500 . But from the very first equation we set up (D + T = 100) we can subtract T from both sides to get D = 100 - T. This means that we can substitute 100 - T for D in the "money" equation. When we make this substitution, the money equation becomes: . 140*(100 - T) + 430T = 28500 . Doing the multiplication on the left side results in: . 14000 - 140T + 430T = 28500 . If you combine the two terms containing T on the left side you get 430T - 140T = 290T. This makes the equation become: . 14000 + 290T = 28500 . Then get rid of the 14000 on the left side by subtracting 14000 from both sides to reduce the equation to: . 290T = 14,500 . Solve for T by dividing both sides by 290 and you have: . T = 14500/290 = 50 . This says that there are 50 cartons of trucks. Since the total number of cartons is 100, that means that there also must be 50 cartons of dolls. . Check using the money relationship. 50 cartons of dolls at \$140 per carton multiplies out to be \$7000. And 50 cartons of trucks at \$430 per carton multiplies out to be \$21500. Adding these two amounts results in \$7000 + \$21500 = \$28500 and that total is what the problem says it should be. Therefore, the answer of 50 cartons of dolls and 50 cartons of trucks is correct. Hope this helps you to understand the problem and how to get the answer. .
 expressions/105567: This question is from textbook Intermediate Algebra Anybody out there who can help with eveluating expressions?? Ineed to evaluate the following expression; log g 1? Thanks a Million.1 solutions Answer 76843 by bucky(2189)   on 2007-10-22 09:28:44 (Show Source): You can put this solution on YOUR website!Given: . . This means that y is the value of . The answer to this problem is that y is equal to zero, so you can say that is equal to zero. Notice that you are saying that the log of 1 is equal to zero regardless of what the base is. . Let's see why. We can do this by translating from the logarithmic form to its equivalent exponential form. This translation says that: . has an equivalent exponential form . By comparing the log form of the translation to the log equation you were given in this problem, you can see that: . and . Then you can see that by substituting the right side of these three relationships into the exponential form, that the exponential form of the given problem is: . . Now it's time to recall that if any base number is raised to the zero power, the answer is 1. Since we need 1 as the answer, we need to raise g to the zero power. So y = 0 and this means that equals zero. . Hope this helps you to understand this problem and how to get the answer. .
 expressions/105566: This question is from textbook Intermediate Algebra Goodmorning everyone, can some one help me to evaluate this expression; log(subscript 8)8^9??? Thanks Ya'll!1 solutions Answer 76839 by bucky(2189)   on 2007-10-22 09:09:41 (Show Source): You can put this solution on YOUR website!Given: . . Note that if you solve for y, the answer is equal to . One way to solve this equation is to translate it to the exponential form of the logarithm. The translation uses the relationship that: . is equivalent to the exponential form . By comparing the log form in the translation to the log form you were given in this problem you can see that: . and . Substituting the values on the right side for b, A, and y in the exponential form you get: . . Looking at this equation, you can see that to make the left side equal to the right side the exponents have to be equal. In other words, y has to be 9 for the two sides to be equal. And since y is equal to , then you can say that equals 9. . Hope this helps you to understand the problem. .
 logarithm/105570: This question is from textbook Intermediate Algebra I need someone's help. I need to find the value of x from logx169/121= 2?? Can someone please help me out on this?? Thank You!1 solutions Answer 76830 by bucky(2189)   on 2007-10-22 08:47:14 (Show Source): You can put this solution on YOUR website!You are given the equation: . . To solve for x you can convert the given logarithmic equation to its equivalent exponential form using the following translation format: . is equivalent to the exponential form . Note that by comparing the given equation to the translation format you can see from the positions in the equations that: . , and . Substituting values for b, A, and y into the exponential form results in: . . You can then solve for x by taking the square root of both sides to get: . . So the answer to your problem is . It's a good idea to become very familiar with switching back and forth between the logarithmic and exponential forms. This switching is often used in solving logarithmic equations. . Hope this helps you to understand this problem.
 Graphs/105571: Find the slope of the line passing through the points (-9,3) and (4,-2)1 solutions Answer 76818 by bucky(2189)   on 2007-10-22 07:12:19 (Show Source): You can put this solution on YOUR website!Given the points (-9,3) and (4,-2) . To find the slope M of the line joining these two points you can use the formula: . . If you let the point {-9, 3) be the first point ... meaning its values of x and y are and ... then the second point is (4, -2) which means that its x and y values are and respectively. Then just plug these values into their locations in the formula and you get: . . So the slope is . You might be able to speed up the process and to do it in your head by just looking at the points (-9, 3) and (4, -2) and say to yourself: . To go from the x value of the first point to the x value of the second point, I would go from -9 to +4. This means that on the x-axis I would move from the left to the right 13 numbers or +13. At the same time, to go from the y value of the first point to the y-value of the second point I would go from +3 to -2 in the y-direction. Since I am moving down the change is negative, and the number of units to get from +3 to -2 is 5, so the change is -5 units. This means the slope is -5 divided by +13 which again gives the answer . Hope this helps you to understand the problem a little better.
 Age_Word_Problems/105576: Rex is 30 years younger than his father. His mother is three times as old as her son. Find Rex's age if his father is two years older than his mother?1 solutions Answer 76817 by bucky(2189)   on 2007-10-22 06:51:53 (Show Source): You can put this solution on YOUR website!See answer 76816 to problem 105578. Problem 105578 is the same as this problem (problem 105576).
 Age_Word_Problems/105578: Rex is 30 years younger than his father. His mother is three times as old as her son. Find Rex's age if his father is two years older than his mother?1 solutions Answer 76816 by bucky(2189)   on 2007-10-22 06:44:50 (Show Source): You can put this solution on YOUR website!Let the age of Rex's father be F. . Then Rex's age (30 years younger than his father) is F - 30. . Rex's mother is 3 times as old as Rex. So her age is 3 times Rex's age or 3(F - 30). . Rex's father's age (which is F) is two years more than his mother's age (which is 3(F - 30). So if you take 2 years from his father's age, the result will equal his mother's age. In equation form this is: . F - 2 = 3(F - 30) . Multiply out the right side and you have: . F - 2 = 3F - 90 . Get rid of the 2 on the left side by adding 2 to both sides: . F = 3F - 88 . Eliminate the 3F on the left side by subtracting 3F from both sides to make the equation become: . -2F = -88 . Solve for F by dividing both sides by -2 and you have: . F = -88/-2 = 44 . So Rex's father is 44 years old. Since Rex is 30 years younger than that, Rex is 44 - 30 or 14 years old. And since Rex's mothers age is 3 times Rex's age, his mother is 3 times 14 or 42 years old. Notice that Rex's father is two years older than Rex's mother, so the problem checks. . In summary ... Rex's age is 14. . Hope this helps you to understand the problem. .
 Polynomials-and-rational-expressions/105574: Tutor please help me with this question 8(1+1/2)^2 +22(1+1/t)+15=0 Thank you! 1 solutions Answer 76814 by bucky(2189)   on 2007-10-22 06:29:16 (Show Source): You can put this solution on YOUR website!Given: . . Inside the first set of parentheses, add the which is the same as to get . Then square that quantity to get . This reduces the problem to: . . Multiplying the 8 times 2.25 results in 18 and the problem then is simplified to: . . Next, multiply the 22 times each of the terms in the remaining set of parentheses. The 22 times the 1 is 22 and the 22 times the is . This makes the equation become: . . Then combine the 18 with the +22 and the +15 to get the sum 55. The equation is then: . . By subtracting 55 from both sides you separate the two terms and make the equation: . . Multiply both sides of this equation by t to eliminate the denominator and make the equation: . . Solve for t by dividing both sides by -55 and you have: . . Notice that both the numerator and the denominator have a common factor of 11. So you can simplify the answer by dividing both by 11 to get: . . Hope this helps you to understand the problem.
 Polynomials-and-rational-expressions/105029: Perform the indicated calculations. Write the result in scientific notation. 4.5 * 10^12/1.5 * 10^7 Thank you for your help..1 solutions Answer 76453 by bucky(2189)   on 2007-10-18 22:39:06 (Show Source): You can put this solution on YOUR website!Given: . . The numerator has two factors ... and . The denominator also has two factors ... and . The way to do this problem is to divide the factor in the denominator into the factor in the numerator. The result of this division is . . Then divide the factor in the denominator into the factor in the numerator. Since these two factors both have the base the division just involves subtracting the exponent of the denominator from the exponent of the numerator. The result is: . . . Combining these two results leads to the answer of and this is in scientific notation. . Hope this helps you to see how to work the problem. .
 Human-and-algebraic-language/103051: WRITE THE EQUATION THAT REPRESENTS EACH STATEMENT; WHEN A NUMBER N IS DIVIDED BY SEVEN THE RESULT IS NEGATIVE TWO1 solutions Answer 74960 by bucky(2189)   on 2007-10-08 22:12:41 (Show Source): You can put this solution on YOUR website!N divided by 7 is: . . This is equal to negative 2 ... which is -2. . Putting this all together results in the equation: . . Hope this helps you to understand the problem.
 Miscellaneous_Word_Problems/103086: A five foot six inch tall woman casts a twelve foot long shadow. at the sam time a tree casts a 45 foot long shadow. How tall is the tree?1 solutions Answer 74954 by bucky(2189)   on 2007-10-08 22:02:56 (Show Source): You can put this solution on YOUR website!You can solve this problem using proportions that are set up as follows: . . The height of the woman is 5 ft 6 inches and this is 5.5 feet. Her shadow is 12 ft long. Substituting these values results in: . . The shadow of the tree is 45 feet. Substitute this and you get: . . You can get rid of the denominator of 45 by multiplying both sides of the equation by 45. When you do that the equation becomes: . . You are now left with the equation: . . Solve for the Height of the tree by multiplying out the numerator of 45*5.5 and dividing that product by 12 to get: . . The tree is 20.625 feet tall. This converts to 20 ft 7.5 inches. . Hope this helps you to understand the problem a little better ... .
 Geometry_Word_Problems/102929: Pls help me with this. It's really hard!! A farmer wants to make a rectangular garden of 7500 square meters, and she has 250 meters of fencing material. There is already a fence on one side. What dimensions should her garden have? I really need some explanation here. Pls help me. thanks. 1 solutions Answer 74854 by bucky(2189)   on 2007-10-08 09:14:53 (Show Source): You can put this solution on YOUR website!Let L represent the length of the rectangle and W represent the width of the rectangle. . The area of the rectangle is to be 7500 square meters. . The area of a rectangle equals the product of the length L and the width W or: . A = L * W . Now substitute 7500 for the area to change the equation to: . 7500 = L * W . The 250 meters of fencing is a little tricky. This fencing will only be needed on 3 sides of the rectangle because the 4th side is an existing fence. So we can say that the fencing material will be used for the Length and the two Widths to be installed, and the existing fence will be used for the other Length. So the 250 meters of fence will equal L plus 2W. In equation form this is: . 250 = L + 2W . Now return to the area equation. We can solve it for W by dividing both sides by L to get: . W = 7500/L . Substitute the right side of this for W in the fence length equation to get: . 250 = L + 2W = L + 2(7500/L) = L + 15000/L . So start with . 250 = L + 15000/L . You can get rid of the denominator by multiplying both sides of this equation (all terms) by L to make the equation become: . 250L = L^2 + 15000 . To get this in "standard form" for solving, make the left side equal zero by subtracting 250L from both sides to change the equation to: . 0 = L^2 - 250L + 15000 . A little more standard form is obtained by simply transposing sides to get: . L^2 - 250L + 15000 = 0 . The left side of this equation can be factored to: . (L - 150)(L - 100) = 0 . This equation will be true if either of the factors equals zero ... because a multiplication by zero on the left side will make the left side of the equation equal the zero on the right side. . Setting each factor equal to zero means: . L - 150 = 0 or L = 150 . and L - 100 = 0 or L = 100 . Let's first choose L = 150 meters. . That leaves 100 meters of fencing material (250 meters - 150 meters) to use in making the 2 widths. So each of the widtha is 100 meters divided by 2 or 50 meters. . So this solution is to make the rectangle 150 meters by 50 meters which multiplies out to give an area of 7500 square meters and use the 250 meters of fence as 2 widths of 50 meters each and a length of 150 meters. 150 meters of the existing fence is used for the 4th side . How about our other potential solution of L = 100 meters? That means that if you cut 100 meters of fence for the length, you have 150 meters left over for the 2 widths. So each width is 75 meters. This makes the dimensions of the garden 100 by 75 meters. This means that the area is 100 times 75 which is 7500 square meters. And the 100 meters plus 75 meters plus 75 meters for the sides adds up to the 250 meters of fencing. The existing fencing can supply the missing 100 meter length. . So there are two answers to this problem. One is to make the rectangle 50 by 150 meters, using the existing fence to provide one of the 150 meter sides. The other is to make the rectangle 75 by 100 meters, using the existing fence to provide one of the 100 meter sides. . Hope this helps you to understand the problem. .
 Equations/102926: The length of a rectangle is 2 in. more than twice its width. If the perimeter of the rectangle is 34 in., find the dimensions of the rectangle.1 solutions Answer 74849 by bucky(2189)   on 2007-10-08 08:09:53 (Show Source): You can put this solution on YOUR website!Let L represent the length and W represent the width. . You are told that L equals 2 inches more than 2W. In equation form this is: . L = 2W + 2 . Next you are told that the perimeter (P) is 34 inches. The perimeter is found by adding all the sides. In other words: . P = L + W + L + W . By combining like terms on the right side this can be simplified to: . P = 2L + 2W . Substitute 34 for the perimeter to get: . 34 = 2L + 2W . But since L = 2W + 2 we can substitute 2W + 2 for L in the perimeter equation to get: . 34 = 2(2W + 2) + 2W . Multiply out the right side by finding the product of 2 and each of the terms inside the parentheses. When you do, the equation becomes: . 34 = 4W + 4 + 2W . Get rid of the 4 on the right side by subtracting 4 from both sides to get: . 30 = 4W + 2W . Combine terms on the right side to reduce the equation to: . 30 = 6W . Solve for W by dividing both sides by 6 to get: . 5 = W . So we know the width is 5 inches. But our first equation tells us that the Length is twice the width plus 2 inches ... or 2 times 5 and then add 2. This makes the length 12 inches. . In summary, the width is 5 inches and the length is 12 inches. . Check by adding the 4 sides: 5 + 12 + 5 + 12 to get 34 inches for the perimeter, just as it should be. . Hope this helps you to understand the problem and how to solve. .
 Linear-equations/102878: what is the slope of the line connecting the two points (-7,5) and (2,-3) 1 solutions Answer 74833 by bucky(2189)   on 2007-10-07 22:34:36 (Show Source): You can put this solution on YOUR website!Let's define Point 1 as (-7,5) and Point 2 as (2, -3). . Then you can say that and . Similarly you can say that and . Then the equation for the slope (M) is: . . Now all you have to do is substitute the values identified above to get: . . So the slope is . Hope this helps you to see your way through the problem. .
 Quadratic_Equations/102876: Tutor, I was hoping you could help me with this problem. I can not seem to get it to check out and I am not sure where I am making the mistake. Solve for X x^2 + 4x + 4 = 7 I know I have the factor the left-hand side but I must be doing something wrong because it is not working out for me. Any help would be great. Thank you in advance for your time. 1 solutions Answer 74828 by bucky(2189)   on 2007-10-07 22:00:03 (Show Source): You can put this solution on YOUR website!If you factor the left side you get: . . Then if you take the square root of both sides you get: . . Note the right side has a plus and minus sign. If you now subtract 2 from both sides to get rid of the 2 on the left side you get: . . Hope this helps you to find the source of your difficulty. .