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# Recent problems solved by 'glabow'

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 Expressions-with-variables/28690: 4X+7Y=32 4X+2Y=641 solutions Answer 15641 by glabow(165)   on 2006-02-27 09:12:08 (Show Source): You can put this solution on YOUR website!To solve equations of this kind you have to think of the variable expession 4X (or whatever) as a number. With an equation you can subtract or add a number to both sides of the equation and it still remains true that the two sides are equal. If you saw 2 + 7 = 9 you know you could subtract 2 from each side and end up with 7 = 7. Well, instead of 2 you have something like 4X. That would be 4X + 7Y = 32. Let's subtract the number 7Y from both sides. You get 4X + 7Y - 7Y = 32 - 7Y. This simplifies to 4X + 2Y = 32 - 7Y. Subtract the number 2Y from both sides, giving 4X = 32 - 5Y. Aha! you now know what 4X equals. Since it's the same X and Y in the second equation, which is 4X + 2Y = 64, you can subtract 4X from both sides of the second equation. But here's the trick ... you use 4X when you subtract from the left side of the equation and you use 32 - 5Y when you subtract from the right side of the equation. They are equal, so you can do this. You are subtracting the same number from each side of the second equation. Doing this you get 4X - 4X = 64 - 32 + 5Y. And that simplifies to 0 = 32 + 5Y. Or, rearranging, 5Y = -32. If 5 times Y equals -32, then 1 times Y must equal -32/5. [Do you see why?] Now you can replace Y with its value. 4X + 2Y = 64 becomes . This is 4X -64/5 = 64. Dividing both sides by 4 gives X - 16/5 = 16. And X = 16 + 16/5, which is X = 96/5. Checking . Right! And . Right! Of course there is a short cut, once you understand where this came from. You can subtract the entire left side of the second equation from the left side of the first equation, and the entire right side of the second equation from the right side of the first equation. You get 4X + 7Y - (4X + 2Y) = 32 - 64 4X + 7Y - 4X - 2Y = 32 - 64 5Y = -32 Y = -32/5 This is the same thing, but just hides all the rules being applied to get there. Practice will make this easy.
 Travel_Word_Problems/21642: Can someone help with this problem! I can't seem to figure it out and my parents are of no help. The rail line between two cities consist of two segments, one 96km longer than the other. A passenger train averages 60km/h over the shorter segment, 120 km/h over the longer, and 100 km/h for the entire trip. How far apart are the cities? 1 solutions Answer 10443 by glabow(165)   on 2005-12-05 08:13:24 (Show Source): You can put this solution on YOUR website!What you need to know is that the time it takes to travel over a distance is given by . This problem has three distances that are given as a shorter distance, a longer distance that is the short distance plus 96, and the total distance which (of course) is their sum. Let x be the shorter distance. The time it took to travel over x was . The time it took to travel over x + 96 was . The time it took to travel over the entire distance was . Of course, the time for the total distance must equal the sum of the times for each segment, so You solve this for x. [get common denominator] [simplify] [multiply both sides by both denominators] [simplify] [divide both sides by 20] [divide both sides by 3] So, x+96= 128, and the total distance is 160. Check by calculating the time taken for each segment and for the total: 32/60 = .5333333333... 128/120 = 1.06666666... 160/100 = 1.6 .5333333... + 1.06666666... = 1.6 [think of it as .5 and 1/3 of a tenth plus 1.0 and 2/3 of a tenth, which is 1.5 plus a whole tenth]
 Linear-equations/21649: please help me solve this question " whats the slope-intercept of a line with the slope 5 and the y- intercept -8?"1 solutions Answer 10439 by glabow(165)   on 2005-12-05 01:47:50 (Show Source): You can put this solution on YOUR website!See the answer to 21648.
 Linear-equations/21648: please help me solve " what the slope-intercept form of of the equation of a line with slope 5 and y-intercept -8?"1 solutions Answer 10438 by glabow(165)   on 2005-12-05 01:45:14 (Show Source): You can put this solution on YOUR website!The slope-intercept form for the equation of a line contains two pieces of information: the slope and the y-intercept. The slope tells how y changes when there is a change in x. For example, a slope of 3 means that for each change in x of 1 unit, y changes 3 units. A slope of -3 would mean that for each change in x of 1 unit, y changes -3 units. This is written algebraicly as y = mx, where m is the slope. y = 3x is the equation of a line that has a slope of 3. Look at a little table for various values of x: y.....x 3.....1 6.....2 9.....3 Each time x changes by 1 (1 to 2, 2 to 3), y changed by 3 (3 to 6, 6 to 9). This is because y is always 3 times whatever x is. Next, the y-intercept tells at what point the line crosses the y axis. That is, what is the value of y when x is zero? Well, since this is a single fixed place on the y-axis the equation of y = a gives that information. If the line crosses the y-axis at 2, then the equation would be y = 2. If it crosses at -7, then the equation would be y = -7. The combination of these two pieces of information gives an equation of . For the information you had (slope of 5 and intercept of -8) the equation is y = 5x - 8. Simple, no?
 Systems-of-equations/19967: m + In Lewis Carroll's Through The Looking Glass, Tweedledum says to Tweedledee, "The sum of your weight and twice mine is 361 pounds." Then Tweedledee says to Tweedledum, "Contrariwise, the sum of your weight and twice mine is 362 pounds." Find the weight of Tweedledum and Tweedledee.1 solutions Answer 9632 by glabow(165)   on 2005-11-17 07:38:51 (Show Source): You can put this solution on YOUR website!Actually, they said no such thing! But if they had..... Let's use two letters for the unknown weights of Tweedledum and Tweedledee. To keep them straight, use the LAST letter of their names: m and e. So, e + 2m = 361 This is the first statement by Tweedledum: "your weight" [e] plust twice MY weight [2m] is 361. The second statement, likewise, is rendered: m + 2e = 362 Now just solve. e + 2m = 361 2e + m = 362 2e + 4m = 722 [multiplying the first equation by 2] 3m = 360 [subtracting the second equation from the third one] m = 120 [dividing both sides by 3] e + 240 = 361 [substituting the value of m in the first equation] e = 121 [solving for e] Checking: 121 + 240 = 361 [first equation] 242 + 120 = 362 [second equation] They don't seem very fat!
 Linear-systems/19970: elimination method 3x+2y=-24; 4x+3y=18 5x-4y=-114; 2x+6y=-571 solutions Answer 9631 by glabow(165)   on 2005-11-17 07:24:00 (Show Source): You can put this solution on YOUR website!I assume your equation pairs read across. Let's take the first one. Elimination means eliminating one of the unknown variables by getting a common coefficient for it in both equations and then subtracting the two sets of equals from each other. [Remember: equals subtracted from equals gives equals.] We have 3x + 2y = -24 4x + 3y = 18 We can make the y terms have the same coefficient by multiplying the first equation by 3 and multiplying the second equation by 2. That will make both equations have 6y as the y-term. This gives 9x + 6y = -72 8x + 6y = 36 Now we subtract the second equation from the first. This leaves x = -108 Since we now know x, we can put that value into either of the original equations and solve for y. 3(-108) + 2y = -24 -324 + 2y = -24 2y = -24 +324 = 300 y = 150 Checking with x=-108 and y=150 shows this to be correct. Now you do the second set. [Hint: multiply the first equation by 6 and the second one by 4 and both equations will have 24 for the coefficient of y. In this case one will be positive and the other negative so you can ADD the equations together to get rid of the y-term.]
 Rational-functions/19972: 5/C - C/C-7 = ? How do you solve this type of equation?1 solutions Answer 9630 by glabow(165)   on 2005-11-17 07:08:47 (Show Source): You can put this solution on YOUR website!Like all algebraic expressions you can only "solve" it when you know a value for the unknown, in this case C. You do not have an equation; you have an expression. You CAN, however, simplify an expression. In this case it doesn't help much since the simplification results in and that's it!
 Length-and-distance/19976: What is -8x+-(5+3x)-21 solutions Answer 9629 by glabow(165)   on 2005-11-17 06:58:51 (Show Source): You can put this solution on YOUR website!This is an expression, meaning it is a sequence of operations and operands. It is algebraic since it contains a number that is unknown and is represented by x. All the other operands (numbers in the expression) can be used in the operations to find out what they are equal to. For example, 3 + 5 has two operands (3 and 5) and one operation ( + ) and the expression 3 + 5 is equal to 8. You have -8x + -(5+3x) - 2 -8x means the negative of 8 times x (whatever number x is, which we don't know!) -(5+3x) means the neagtive of the quantity 5+3x, and you know what 3x means, right? and 2 the first number [-8x] is added to the second number [-(5+3x) and then the third number [2] is subtracted from that sum. This can be simplified by applying the negative of the quantity (5+3x) giving -5 -+3x. It is usual to write a negative addition as a subtraction, so this becomes -5 - 3x. The whole expression is now -8x +-5 - 3x - 2 Again, the addition of a negative is usually written as a subtraction, so: -8x - 5 - 3x - 2 Now terms can be grouped and rearranged. -8x - 3x - 5 - 2 And arithmetic can be done, giving -11x -7 That's the simplified expression which will have the same value as the original. Of course, the value of the expression is unknown until you know what value x represents. Example: if x=3, the expression becomes -11(3) -7 == -33 - 7 = -40. Hope that helps.
 real-numbers/19101: Tripling the greater of two consecutive even integers gives the same result as subtracting 10 from the lesser even integer. What are the integers? I need help on everything in this questing I need to know how to do it and show my work! Please help1 solutions Answer 9171 by glabow(165)   on 2005-11-09 19:34:49 (Show Source): You can put this solution on YOUR website!First, what are two consecutive even integers. If you let x be the first one, what would the next even integer be? If you said x+2 you would be right. So, the smaller of the two is x, the larger is x+2. Now you know that 3 times the larger one is equal to the smaller one less 10. 3 times the larger one is written algebraicly like this: 3(x+2). The smaller one less 10 is written like this: x - 10. You know 3(x+2) = x-10. Can you solve this equation for x? 3x+6 = x-10 [distributive rule] 2x+6=-10 [subtract x from both sides] 2x=-16 [subtract 6 from both sides] x=-8 [divide both sides by 2] x+2=-6 So three times the greater number (-6) is equal to the smaller number (-8) minus 10. [Checking: 3(-6)=-18. -8-10=-18.]
 Expressions-with-variables/16722: Please help me solve this equation. 8p - 5(p + 3) = (7p - 1)31 solutions Answer 8156 by glabow(165)   on 2005-10-23 20:24:38 (Show Source): You can put this solution on YOUR website!This is an example of using the distributive property: a(b+c) = ab + ac. 8p -5(p + 3) = (7p - 1)3 8p - 5p -15 = 21p -3 Now it's a matter of rearranging terms (addition and subtraction are commutative). 3p - 15 = 21p - 3 - 15 + 3 = 21p - 3p [adding 3 to both sides and subtracting 3p from both sides] -12 = 18p p = - 12/18 = - 2/3 HTH
 Distributive-associative-commutative-properties/16726: ok I am having some trouble on "finishing" the product of some Distributive properties ok, for equation -9(3x + 8) I put down the following steps: = -9*3x + -9*8 = -27x + -72 (final answer) I am not sure if this is correct or not. My other question on the same topic is an equation i did like so: 5(x + 1) = 5*x + 5*1 = 5x + 5 (final answer) please let me know if i am doing the distibutive property correctly. 1 solutions Answer 8155 by glabow(165)   on 2005-10-23 20:19:02 (Show Source): You can put this solution on YOUR website!Everything you posted looks right. Remember: a(b+c) = ab + ac. That's it.
 Matrices-and-determiminant/14571: Help me please! I don't understand what I am suppose to do. Evaluate the determinant of the matrix. items in a bracket. 3 -2 0 -1 5 3 2 7 11 solutions Answer 7314 by glabow(165)   on 2005-10-10 09:32:49 (Show Source): You can put this solution on YOUR website!The underlying meaning of a determinant is complex and cannot be covered here. It is a useful value to know when working with matrices. There are a couple of ways of calculating the determinant of a matrix, and for a 3 x 3 there is a relatively easy procedure. I'm surpised you haven't been given it (if you haven't) but I'll try it here. You first get the products of all the left-to-right diagonals of the matrix and add them up. You then get the products of all the right-to-left diagonals of the matrix and add them up. You then subtract the right-to-left sum from the left-to-right sum. The diagonals are all referenced from top to bottom. You can see how this is a very poor way to describe the procedure. Let's make it mathematical. We will reference the elements of the matrix with the following notation: will mean the element in row x and column y. In a 3 x 3 matrix the upper left element will be and the lower right element will be . Get it? Okay. The left-to-right diagonals are as follows: , , ; , , ; and , , Whew! For the matix values you gave, this amounts to (3 x 5 x 1), (-2 x 3 x 2), and (0 x -1 x 7). Do the multiplications and add them up. 15+(-12)+0 = 3. This is the value of the products of all the left-to-right diagonals. The right-to-left diagonals are as follows: , , ; , , ; and , , For the matrix values you gave, this amounts to (0 x 5 x 2), (-2 x -1 x 1), and (3 x 3 x 7). Do the multiplications and add them up. 0+2+63 = 65. This is the value of the products of all the right-to-left diagonals. Subtract the right-to-left from the left-to-right. Determinant = 3 - 65 = -62 CHECK MY WORK!!!
 Travel_Word_Problems/14579: two cares leave st.louis. one travels north and one travles south. the care traveling north is moving 10mph faster than the care traveling south. after three hours the cars are 354 miles apart. how fast is each car traveling? 1 solutions Answer 7312 by glabow(165)   on 2005-10-10 09:08:19 (Show Source): You can put this solution on YOUR website!Ah, if I could get rid of my cares so easily! [watch those typos] It really doesn't matter if they are going North, South, or any other direction. The question boils down to: two cars travel. One goes at a speed of ? miles per hour. Let's use the letter x for that unknown speed. The other one goes x+10 miles per hour. [Do you see why?] After three hours of travel, the total distance they have traveled is 354 miles. The slower car goes 3 hours times x miles per hour, which is 3x miles. The faster car goes 3 hours times x+10 miles per hour, which is 3(x+10) miles. The total distance they travel is 3x + 3(x+10) = 354 miles. This is easy to solve. 3x + 3x + 30 = 354 [simplifying the expression] 6x + 30 = 354 [more simplifying] 6x = 324 [subtracting 30 from both sides] x = 54 [dividing both sides by 6] x+10 = 64 Checking: 3(54) + 3(64) = 162 + 192 = 354
 Graphs/14610: Please help with this inequality: 8.9k is greater or less than 17.8 thanks!1 solutions Answer 7311 by glabow(165)   on 2005-10-10 08:58:22 (Show Source): You can put this solution on YOUR website!I would like to add that the phrase "greater or less than" is referring to those values of k which make the inequality true. By solving for equality you have a value for which "greater or less than" is NOT true. So, the correct answer is any value EXCEPT k=2.
 Graphs/14606: Please help me find the inequality for: 22is less then or greater to b+221 solutions Answer 7309 by glabow(165)   on 2005-10-10 08:54:57 (Show Source): You can put this solution on YOUR website!Read the answer to 14607 and think it over.
 Graphs/14608: PLease help me with this inequality: n-34 is less than or greater to 16 thanks a lot!1 solutions Answer 7308 by glabow(165)   on 2005-10-10 08:54:23 (Show Source): You can put this solution on YOUR website!Read the answer to 14607 and think it over.
 Graphs/14609: please help with this inequality: x + 3.4 > 5.8 thank you!1 solutions Answer 7307 by glabow(165)   on 2005-10-10 08:53:51 (Show Source): You can put this solution on YOUR website!Read the answer to 14607 and think it over.
 Graphs/14611: Please help me with this inequality:X divided by 2 is less than or greater to 52 Thank you very much!1 solutions Answer 7306 by glabow(165)   on 2005-10-10 08:53:21 (Show Source): You can put this solution on YOUR website!Read the answer to 14607 and think it over.
 Graphs/14607: Please help me with this inequality: 56<14t thank you!1 solutions Answer 7304 by glabow(165)   on 2005-10-10 08:51:02 (Show Source): You can put this solution on YOUR website!What do you want to know? I assume you wonder what values of t make the statement true! Try the following: If t=4, then 14t=56. What happens if t gets smaller, say 3.999. 14t will then be smaller than 56. What happens if t gets larger, say 4.00001. 14t will then be larger than 56. Any number larger than 4 would then make the inequality 56<14t true. So an "answer" would be: 56<14t for all values of t>4. I suspect that is what you were looking for. NOTE: while this is all intuitive, you can be more formal. 56 < 14t 4 < t [dividing both sides by 14] This says the same thing. The only thing to remember about inequalities is that you can add, subtract, multiply and divide by equals on both side BUT MUST CHANGE THE DIRECTION OF THE INEQUALITY if you divide or multiply by a negative number. 3 > 2 [true] 6 > 4 [multiplying both sides by 2] BUT -6 < -4 [multiplying both sides by -2] Play with this a bit with simple numbers (and draw them on a number line) until you see how this works.
 Rectangles/14604: A rectangular yard is enclosed with 480 ft of fencing material; that is, its perimeter is 480 feet. Its length is twice its width. What is its length? 1 solutions Answer 7302 by glabow(165)   on 2005-10-10 08:42:38 (Show Source): You can put this solution on YOUR website!Okay, picture the rectangle. The length of one side is x, i.e. a value we don't yet know so we use a letter to indicate that value. The other side is twice that length. So, the other side is 2x. When you add up the sides (x + 2x + x + 2x) you get a value of 480. Putting this into an algebraic expression (instead of words) gives . This can be solved easily. 6x = 480 x = 80 [dividing both sides by 6] That is the length of one side. 2x = 160 [multiplying both sides by 2] That is the length of the other side.