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 Miscellaneous_Word_Problems/203188: How many points in the first quadrant satisfy both of the following conditions: (i) the x and y coordinates are positive integers and (ii) x + y ≤ 5?1 solutions Answer 153301 by jim_thompson5910(28595)   on 2009-07-15 17:32:35 (Show Source): You can put this solution on YOUR website!Since "x" and "y" are both positive integers (ie positive whole numbers 1, 2, 3, 4, etc...) just ask yourself: "what two numbers add to another number that is either less than or equal to 5?" So for instance, 2+2=4 which is less than or equal to 5. Also, 1+2=3 which is also less than or equal to 5. Furthermore, 2+3=5 which is less than or equal to 5. These values can be thought of as the ordered pairs (2,2), (1,2), and (2,3) Here are all of the possible ordered pairs (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (2, 0), (2, 1), (2, 2), (2, 3), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (5, 0) So for example, the ordered pair (1,4) means that 1+4=5 which is less than or equal to 5.
 Square-cubic-other-roots/203195: I need help rationalizing.... 9 times the cubic root 2 over/divided by the cubic root 5 minus cubic root 2. Please help! Soory my keyboard will not allow me to type the equation. 1 solutions Answer 153300 by jim_thompson5910(28595)   on 2009-07-15 17:23:25 (Show Source): You can put this solution on YOUR website! Start with the given expression. Let and Replace each with "x" and each with "y" Now this is where you have to get a bit creative. It's very helpful to remember that . Why is this important? Because if you cube a cube root, they will both cancel out (leaving the stuff inside as a remainder). So if we multiply both numerator and denominator by , we get: Use the identity given above Plug in and Cube the cube roots (to eliminate them) Subtract Reduce Combine the roots. Multiply Use the identity Square 5 to get 25 and square 2 to get 4 Distribute Combine the roots. Multiply ================================================================== Answer: So This can be verified (as I have verified it myself) with a calculator.
 Trigonometry-basics/203109: i was wondering how you do these problems: The directions are to find the exact value of each expression. 6 cos 3π/4+ 2tan(-π/3) and sec ( -π/3)- cot (-5π/4) please i am struggling to do this work. I would appreciate if you would explain in detail. Thank you in advance.1 solutions Answer 153249 by jim_thompson5910(28595)   on 2009-07-14 23:48:19 (Show Source): You can put this solution on YOUR website!Ok, first off let's bring up the unit circle This is a very handy tool. For each given angle (located on the spokes of the wheel), there is a point that lies on the circle. It turns out that the x coordinate of the point refers to the cosine of that given angle. Likewise, the y coordinate of the point refers to the sine of that angle. For example, the cosine of is since the x coordinate of this point is . Also, the sine of is since the y coordinate of this point is . You can use the unit circle to evaluate the sine/cosine of any given angle shown. Finally, remember that we have the following identities: I hope this will refresh something or help you get started. If this didn't help, please let me know or repost.
 Money_Word_Problems/203103: Debbie plans on investing enough money so that she will have $2000 at the end of two years. How much money wll she have to invest now if her money will grow according to the followning formula: 2000=P(1.025)^4? When I work this problem I seem to come up with 0=0 2000=P(1.025)^4 1.025^4=1.103812891 I don't understand.1 solutions Answer 153235 by jim_thompson5910(28595) on 2009-07-14 19:56:30 (Show Source): You can put this solution on YOUR website!I'm assuming that you have the correct formula... Start with the given equation. Raise 1.025 to the 4th power to get 1.103812 Divide both sides by 1.103812 to isolate "P". Divide Rearrange the equation and round to the nearest hundredth (ie to the nearest penny). So she needs to invest$1,811.90
 Radicals/203101: This question is from textbook {{cube root of 108x^3y^7/2y^3}}1 solutions Answer 153233 by jim_thompson5910(28595)   on 2009-07-14 19:38:04 (Show Source): You can put this solution on YOUR website! Start with the given expression. Reduce the coefficients. Divide the common variable terms by subtracting the exponents. Subtract Factor 54 into 27*2 Factor into Break up the root. Take the cube root of 27 to get 3 Take the cube root of to get "x" Take the cube root of to get "y" Rearrange the terms and combine the roots. So where
 Complex_Numbers/203100: This question is from textbook Imaginary numbers x^2-2x+12,x=1-2i1 solutions Answer 153231 by jim_thompson5910(28595)   on 2009-07-14 19:31:26 (Show Source): You can put this solution on YOUR website! Start with the given expression. Plug in FOIL Distribute Combine like terms. Plug in Multiply Subtract So This means that when
 Graphs/203102: #1 What is the slope of the line containing the points (-1,4) and (3,2)? a. -2 b. 1/2 c. -1/2 d. 2 -------------------------------------------------------------------------------- Question 2 (Multiple Choice Worth 1 points) Choose the best description for graphing the point (3,-4). a. Go to the left three and up four. b. Go to the right three and up four. c. Go to the right three and down four. d. Go to the left three and down four. -------------------------------------------------------------------------------- Question 3 (Multiple Choice Worth 1 points) Put the following equation in slope intercept form 2x - 5y = 10. a. x-y = (-2/5)x-2 b. y = (-2/5)x + 2 c. y = -2x + 10 d. y = (2/5)x -2 -------------------------------------------------------------------------------- Question 4 (Multiple Choice Worth 1 points) What is the slope of the line with equation y = (1/2)x -3? a. 1/2 b. -3 c. 2 d. -2 -------------------------------------------------------------------------------- Question 5 (Multiple Choice Worth 1 points) In the equation y=-2x +5, name the slope and the y intercept. a. slope 1/2, y intercept 5 b.slope -2, y intercept 5 c.slope 5, y intercept -2 d. slope 5, y intercept 1/2 -------------------------------------------------------------------------------- Question 6 (Multiple Choice Worth 1 points) What is the y intercept of the equation y= (2/3)x -6? a. 2/3 b. 6 c. -6 d.No y intercept -------------------------------------------------------------------------------- Question 7 (Multiple Choice Worth 1 points) Choose the best directions for graphing the equation y= (1/2)x + 1. a. Graph the point (0,1) and count up 1 and to the right 2 to find another point on the line. b. Graph the point (1,0) and count up 1 and to the right 2 to find another point on the line. c. Graph the point (0,1) and count to the right 1 and up 2 to find another point on the line. d. Graph the point (1,0) and count to the right 1 and up 2 to find another point on the line. -------------------------------------------------------------------------------- Question 8 (Multiple Choice Worth 1 points) What kind of line is created when you graph the equation x=2? a.Horizontal line b.Oblique line c.Vertical line d.No line is created. -------------------------------------------------------------------------------- Question 9 (Multiple Choice Worth 1 points) What is the slope of the line x=2? a.Zero b.Undefined c.2 d.Not enough information. -------------------------------------------------------------------------------- Question 10 (Multiple Choice Worth 1 points) What is the slope of the line y= -3? a.Zero b.Undefined c.-3 d.Not enough information. -------------------------------------------------------------------------------- Question 11 (Multiple Choice Worth 1 points) Choose the equation that is written in standard form. a.y= -3x +4 b.2y = x+8 c.x = 3y-6 d.4x -2y = 10 -------------------------------------------------------------------------------- Question 12 (Multiple Choice Worth 1 points) Find the x intercept of the graph represented by the equation 3x + 4y = 12. a.(0,3) b.(3,0) c.(4,0) d.(0,4) -------------------------------------------------------------------------------- Question 13 (Multiple Choice Worth 1 points) Find the y intercept of the graph represented by the equation 3x + 4y = 12. a.(0,3) b.(3,0) c.(4,0) d.(0,4) -------------------------------------------------------------------------------- Question 14 (Multiple Choice Worth 1 points) Choose ALL the points that are solutions to the equation y= 2x -5. a.(0,-5) b.(1,3) c.(-1,-7) d.(1/2, -4) -------------------------------------------------------------------------------- Question 15 (Multiple Choice Worth 1 points) Choose the equation of the line with a slope of 2/3 and y passing through the point (6,-2). a.y= (2/3)x -6 b.y= (2/3)x -2 c.y= (2/3) x -4 d.y = (3/2) x -6 -------------------------------------------------------------------------------- Question 16 (Multiple Choice Worth 1 points) Choose the equation of the line passing through the points (-2,3) and (-4,1). a.y = x -1 b.y = (-1/3)x + 7/3 c.y= x-4 d.y = x + 5 -------------------------------------------------------------------------------- Question 17 (Multiple Choice Worth 1 points) Choose the equation of the vertical line passing through the point (-2,1). a.x = -2 b.x = 1 c.y = -2 d.y = 1 -------------------------------------------------------------------------------- Question 18 (Multiple Choice Worth 1 points) Choose the equation of the line passing through the points (3,4) and (1,-6). a.y = 5x - 19 b.y = (1/5)x + 15/5 c.y = (1/5)x - 2 d.y = 5x - 11 -------------------------------------------------------------------------------- Question 19 (Multiple Choice Worth 1 points) Choose the ordered pair that is the solution to the following system of equations: y= 2x + 1 and y = x -5. a.(-6, -1) b.(-6,-11) c.(4,9) d.(4,-1) -------------------------------------------------------------------------------- Question 20 (Multiple Choice Worth 1 points) Choose the orderred pair that is the solution to the following system of equations: -3x + y = 4 and -7x + y = 12. a.(-2,-2) b.(-2,2) c.(2,-2) d.(2,2) -------------------------------------------------------------------------------- Question 21 (Multiple Choice Worth 1 points) Choose the ordered pair that is the solution to the following system of equations: 5x - 2y = 3 and 3x + 4y = 7. a.(1,-1) b.(-1,1) c.(-1,-1) d.(1,1) -------------------------------------------------------------------------------- Question 22 (Essay Worth 1 points) Give an ordered pair that is the solution to the equation y = 5x + 2. -------------------------------------------------------------------------------- Question 23 (Multiple Choice Worth 1 points) Choose the equation of the line that is perpendicular to the line shown in the graph and passes through the point shown in the graph. a.y = x - ll b.y= 3x – 11 c.y = -3x - 11 d.y = -x + 11 -------------------------------------------------------------------------------- Question 24 (Multiple Choice Worth 1 points) Choose the equation of the line that is perpendicular to the line shown in the graph and passes through the point shown in the graph. a.y = 2x + 7 b.y = -2x + 10 c.y = 2x - 10 -------------------------------------------------------------------------------- Question 25 (Multiple Choice Worth 1 points) The figure shown below looks like a trapezoid. By definition, a trapezoid is a four-sided figure with at least two parallel sides. Choose the best description below of how to determine whether the figure shown is or is not a trapezoid. a.Find the slope of the line between point (-9,5) and point (-3,8). Then find the slope of the line between point (4,-1) and point (3,5). If the slopes are opposite reciprocals, then the figure is a trapezoid. b.Find the slope of the line between point (-3,8) and point (3,5). Then find the slope of the line between point (-9,5) and point (4,-1). If the slopes are the same, then the figure is a trapezoid. c.Find the slope of the line between point (-9,5) and point (3,5). Then find the slope of the line between point (-3,8) and point (4,-1). If the slopes are the same, then the figure is a trapezoid. d.Find the slope of the line between point (-3,8) and point (3,5). Then find the slope of the line between point (-9,5) and point (4,-1). If the slopes are opposite reciprocals, then the figure is a trapezoid. 1 solutions Answer 153229 by jim_thompson5910(28595)   on 2009-07-14 19:23:05 (Show Source): You can put this solution on YOUR website!We're not here to do your homework. If that's the case, you might as well give us your degree, your job, and any money that you make from it. Not much fun is it? If you are struggling with the problems, please break them up and show that you've attempted to solve them. Also, post the problems that are giving you the most trouble.
 Exponential-and-logarithmic-functions/203087: Find the absolute maximum and absolute minimum values of the function below. If an absolute maximum or minimum does not exist, enter NONE. f(x) = x^2 + 250/x on the open interval (0,infinity ) I know that the absolute max is the answer NONE but I can not figure out the absolute min can someone help please thanks1 solutions Answer 153219 by jim_thompson5910(28595)   on 2009-07-14 16:28:18 (Show Source): You can put this solution on YOUR website!Here are the basics to finding the extrema (ie the max/min) Step 1) Derive the function f(x) to get f'(x) Since this means that or simply Step 2) Set the derivative function f'(x) equal to zero Multiply both sides by and rearrange the equation. Add 250 to both sides. Take the square root of both sides Simplify the square root Evaluate the square root Now because the domain is , this means that we must reject (as it isn't in the domain) So the max/min occurs at the approximate value (or at the endpoints) Now if you graph the function (or evaluate a list of x values), you'll find that there is no max value (so you are correct). So this means that the min occurs when or the min occurs at the endpoints of the interval. Because x=0 is not defined (and infinity is not a number), this means that the min must occur at Step 3) Find the minimum value of the function f(x) From here, simply plug in the x value that generates the min value, which in this case is to get So the absolute min is approximately f(x)=31.6228 and occurs at x=15.8114 Note: if you want to keep things exact, then you can plug in to get the min value: . This would then mean that the absolute min of occurs at
 Pythagorean-theorem/203082: the sum of two numbers is 12. their product is 32. find the numbers. not good at word problem so i am have trouble setting it up. once i get it set up i can work its just braking it down to set it up. please help.1 solutions Answer 153199 by jim_thompson5910(28595)   on 2009-07-14 14:29:05 (Show Source): You can put this solution on YOUR website!Let x = first number y = second number "the sum of two numbers is 12" translates to and "their product is 32" means that So you basically have the system I'll let you take it from there. Let me know if you need more help.
 Linear-systems/203073: -3x+y=3 and 2x-y=-1 what is the solution1 solutions Answer 153190 by jim_thompson5910(28595)   on 2009-07-14 13:27:04 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Start with the first equation. Add to both sides. Rearrange the terms and simplify. ------------------------------------------- Move onto the second equation. Now plug in . Distribute. Combine like terms on the left side. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ------------------------------------------- Since we know that , we can use this to find . Go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. So the solutions are and . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)
 Polynomials-and-rational-expressions/203062: I have to perform the operation of:  k^2+3k-10 3 _________ x ________ 36 k^2-4  I tried to look for simplification by factoring out the first numerator to (k+5)(k-2). But I don't see how I can factor out a (k-2) from the second denominator because there's no k value in the expression. However, the choices for answers indicate that it is cancelled out somehow. Where do I go from here?1 solutions Answer 153184 by jim_thompson5910(28595)   on 2009-07-14 13:01:45 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor the first numerator Factor the first denominator Factor the second denominator (hint: use the difference of squares formula) Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify Distribute So simplifies to In other words,
 Polynomials-and-rational-expressions/203065: (n^2+5n+6/n^2+7n+12)(n^2+9n+20/n^2+11n+30)= ?1 solutions Answer 153181 by jim_thompson5910(28595)   on 2009-07-14 12:37:08 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get . Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words,
 Polynomials-and-rational-expressions/203014: (3x^2 + 4x – 3) + (2x^2 – 3x -1) – (x^2 + x + 7) I just do not get this. I do not understand any algebra at all. It might as well be Chineese. I am in my 5th week of 10 and am failing. I need to pass this class. I need ti understand. The light just won't go on? Please help, Angela1 solutions Answer 153155 by jim_thompson5910(28595)   on 2009-07-13 23:13:04 (Show Source): You can put this solution on YOUR website!When we want to add two variable expressions like "x" and "2x", what we're really doing is adding the coefficients. So x+2x=3x. Remember "2x" is the same as "x+x". So x+2x=x+x+x=3x Start with the given expression. Distribute Group like terms. Combine like terms. So
 Equations/202989: 3-(x/3)=2x/5 This is actually a rational equation but i couldn't find it in the topics.1 solutions Answer 153142 by jim_thompson5910(28595)   on 2009-07-13 19:49:10 (Show Source): You can put this solution on YOUR website! Start with the given equation. Multiply EVERY term by the LCD to clear any fractions. Multiply and simplify. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is which approximates to .
 Graphs/202967: What is an equation of the points on the graph(0,5) and (1,1) ?1 solutions Answer 153106 by jim_thompson5910(28595)   on 2009-07-13 15:27:21 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Note: is the first point . So this means that and . Also, is the second point . So this means that and . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get Reduce So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Distribute Multiply Add 5 to both sides. Combine like terms. Simplify So the equation that goes through the points and is Notice how the graph of goes through the points and . So this visually verifies our answer. Graph of through the points and
 Linear-systems/202943: Explain how to apply elimination in solving a system of equations; apply substitution in solving a system of equations. Demonstrate each technique in solving the system 2x + 7y = 12 5x - 4y = -131 solutions Answer 153104 by jim_thompson5910(28595)   on 2009-07-13 14:38:43 (Show Source): You can put this solution on YOUR website!Start with the given system of equations: Start with the first equation. Subtract from both sides. Divide both sides by to isolate . Rearrange the terms and simplify. ------------------------------------------- Move onto the second equation. Now plug in . Distribute. Multiply both sides by the LCD to clear any fractions. Distribute and multiply. Combine like terms on the left side. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ------------------------------------------- Since we know that , we can use this to find . Go back to the first equation. Plug in . Multiply. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the solutions are and . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)
 Polynomials-and-rational-expressions/202948: Factor: 64x^3+27y^31 solutions Answer 153103 by jim_thompson5910(28595)   on 2009-07-13 14:36:27 (Show Source): You can put this solution on YOUR website! Start with the given expression. Rewrite as . Rewrite as . Now factor by using the sum of cubes formula. Remember the sum of cubes formula is Multiply ----------------------------------- Answer: So factors to . In other words,
 Graphs/202960: Could you please help me out? Find an equation of the line having the given slope and containing the given point. m=7/8, (5,-2) The equation of the line is y = Thanks for your help!!1 solutions Answer 153102 by jim_thompson5910(28595)   on 2009-07-13 14:34:02 (Show Source): You can put this solution on YOUR website! If you want to find the equation of line with a given a slope of which goes through the point (5, -2), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Distribute Multiply and to get Subtract 2 from both sides to isolate y Combine like terms and to get (note: if you need help with combining fractions, check out this solver) ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (5, -2) is: which is now in form where the slope is and the y-intercept is
 Graphs/202961: I need help with this algebra problem... Find an equation of the line containing the given pair of points. (-7,-5) & (-3,-2) in slope intercept form y= Thank you for your help!!1 solutions Answer 153101 by jim_thompson5910(28595)   on 2009-07-13 14:32:00 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Note: is the first point . So this means that and . Also, is the second point . So this means that and . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Rewrite as Rewrite as Distribute Multiply Subtract 5 from both sides. Combine like terms. note: If you need help with fractions, check out this solver. So the equation that goes through the points and is
 Expressions-with-variables/202957: Please help me solve this equation. find f(1) for f(x)=2x^3-3x^2+x-211 solutions Answer 153100 by jim_thompson5910(28595)   on 2009-07-13 14:28:45 (Show Source): You can put this solution on YOUR website!Note: we're not solving the equation, we're simply plugging in a value for "x" and evaluating. Start with the given equation. Plug in . Cube to get . Square to get . Multiply and to get . Multiply and to get . Combine like terms. So the answer is
 Expressions-with-variables/202875: Hi I need help with an area of a trapezoid problem please. The area of a trapezoid is given by A=1/2h(a+b) Solve for a. Not even sure where to start on this one.1 solutions Answer 153033 by jim_thompson5910(28595)   on 2009-07-12 20:54:29 (Show Source): You can put this solution on YOUR website! Start with the given equation. Multiply both sides by 2. Distribute Subtract hb from both sides. Divide both sides by "h" to isolate "a". So the solution is