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# Recent problems solved by 'richwmiller'

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 Systems-of-equations/260288: I have tried 3 times to get the answer correct for the problem: 3/4x-1/2y=8 and 3/2x+1/2y=19 i know i am supposed to find y first to get x and plug it in, so i have to multiply by a common denominator, i have tried 2 and 4, my math teacher gave us the answers so we could try again and again and the answer is x=12 and y=2 but i dont see how they get those answers. Can you help? 1 solutions Answer 191721 by richwmiller(9135)   on 2010-01-24 11:59:02 (Show Source): You can put this solution on YOUR website!3x-2y=32 3x+y=38 -2y-y=-6 -3y=-6 y=2 3x+2=38 3x=36 x=12
 Linear-equations/260287: How do I plot y=-4 on a graph1 solutions Answer 191720 by richwmiller(9135)   on 2010-01-24 11:57:08 (Show Source): You can put this solution on YOUR website!y never changes but x does It is a horizontal line at y=-4 It goes straight across the page. It has 0 slope no slant
 Linear-equations/260285: Determine whether the graphs of the two lines are parallel y+3=6x 8x-y=-31 solutions Answer 191719 by richwmiller(9135)   on 2010-01-24 11:54:17 (Show Source): You can put this solution on YOUR website!get them both in slope form y=mx+b if the m's are the same they are parallel (They are not)
Equations/260263: 2x^3+2x^2-4x over x^2-6x+9 simplify
1 solutions

Answer 191716 by richwmiller(9135)   on 2010-01-24 11:40:51 (Show Source):
You can put this solution on YOUR website!
(2x(x^2+x-2))/(x-3)^2

Below are instructions on how to factor x^2+x-2 and x^2-6x+9
Well worth learning. They are long but easy to follow.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 2 and the last term -2 to get -4. Now we need to ask ourselves: What two numbers multiply to -4 and add to 1? Lets find out by listing all of the possible factors of -4

Factors:

1,2,4,

-1,-2,-4, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -4.

(-1)*(4)=-4

(-2)*(2)=-4

Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1

||||
 First Number | Second Number | Sum 1 | -4 | 1+(-4)=-3 2 | -2 | 2+(-2)=0 -1 | 4 | (-1)+4=3 -2 | 2 | (-2)+2=0

None of these factors add to 1. So the quadratic cannot be factored.

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 1 and the last term 9 to get 9. Now we need to ask ourselves: What two numbers multiply to 9 and add to -6? Lets find out by listing all of the possible factors of 9

Factors:

1,3,9,

-1,-3,-9, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 9.

1*9=9

3*3=9

(-1)*(-9)=9

(-3)*(-3)=9

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6

||||
 First Number | Second Number | Sum 1 | 9 | 1+9=10 3 | 3 | 3+3=6 -1 | -9 | -1+(-9)=-10 -3 | -3 | -3+(-3)=-6

We can see from the table that -3 and -3 add to -6. So the two numbers that multiply to 9 and add to -6 are: -3 and -3

breaks down to this (just replace with the two numbers that multiply to 9 and add to -6, which are: -3 and -3)

Replace with

Group the first two terms together and the last two terms together like this:

Factor a 1x out of the first group and factor a -3 out of the second group.

Now since we have a common term we can combine the two terms.

Combine like terms.
==============================================================================

which can also be written as since the factors repeat themselves

Notice how foils back to our original problem . This verifies our answer.

 Linear-equations/260279: I keep coming up with the wrong answer. solution sub. y=3x-4 y=3x+4 My answer is 3x-4=3x+4 0=0 is a solution -3x =-3x ____________ 0 = 0 +4= +4 _______________ 1 solutions Answer 191714 by richwmiller(9135)   on 2010-01-24 11:23:39 (Show Source): You can put this solution on YOUR website!Your solutions should tell you that there is no solution. Look at the equations. They are in slope intercept form. They have the same slope but different y intercepts They are parallel lines and of course never cross. A solution would be where they cross. but they don't.
 Linear-systems/260278: Solve the system: {-x + y = 2 {2x + y = -1 Please and Thank you!1 solutions Answer 191711 by richwmiller(9135)   on 2010-01-24 11:18:26 (Show Source): You can put this solution on YOUR website!There are many ways to solve these of course. 1)multiply the first by 2 and then add the equations will eliminate x 2)subtract either from the other as they are now will eliminate y 3)solve both for y and then set them equal
 Length-and-distance/260274: Give the coordinates of two points on a coordinate grid that are square root of 10 apart.1 solutions Answer 191709 by richwmiller(9135)   on 2010-01-24 11:14:57 (Show Source): You can put this solution on YOUR website!Working backwards with the distance formula to find the points sqrt(10)^2=10 a^2+b^2=10 1^2+3^2=10 (1-0)^2+(3-0)^2) (0,0) and (3,1) There are of course many more combinations.
 Linear-systems/260273: Solve each system: {y = 2x + 8 {y = 3x - 1 {2x - y = 2 {2x - 2y = 41 solutions Answer 191707 by richwmiller(9135)   on 2010-01-24 11:07:14 (Show Source): You can put this solution on YOUR website!the the first two both =y set them equal to each other and solve for x 2x+8=3x-1 9=x y=27-1 y=26 You are supposed to have only one problem per submission and no similar problems the second one becomes similar by solving both equations for 2x and setting them equal
 Number-Line/260266: On a number line, graph the solutions to the inequality 2x+6<6.1 solutions Answer 191706 by richwmiller(9135)   on 2010-01-24 11:03:38 (Show Source): You can put this solution on YOUR website!2x+6<6 2x<0 x<0 So this will be the whole left side of the graph paper. or on the number line from zero (not included) to the left including all negative numbers for x
 Travel_Word_Problems/260260: Michael drove to a friend's house at a rate of 40mph. He returned by the same route at a rate of 45mph. The driving time for the round trip was 4 hours. What is the distance Michael traveled?1 solutions Answer 191704 by richwmiller(9135)   on 2010-01-24 10:58:33 (Show Source): You can put this solution on YOUR website!40*t=45(4-t) 40t=180-45t 85t=180 17t=36 t=36/17 =2 2/17 40*36/17 4-2 2 /17 32/17*45=40*36/17 total trip 84 12/17 each way 169 7/17 round trip
 Polynomials-and-rational-expressions/260258: (y²-8)(6y²-9y+5)1 solutions Answer 191698 by richwmiller(9135)   on 2010-01-24 10:39:24 (Show Source): You can put this solution on YOUR website!(y²-8)(6y²-9y+5) first do (y²)*(6y²-9y+5) 6y^4-9y^3+5y^2 then do (-8)(6y²-9y+5) remember -1*-1=1 -48y^2+72y-40 then put them altogether and combine variables when possible and add up all constants 6y^4-9y^3+5y^2-48y^2+72y-40 combine 5y^2 and -48y^2=-43y^2 6y^4-9y^3-43y^2+72y-40 and we are finished
 Probability-and-statistics/260208: If a multiple choice test consist of 5 question each with four possible answers of which only one is correct , in how many different ways can a student check one of the question?1 solutions Answer 191697 by richwmiller(9135)   on 2010-01-24 10:27:52 (Show Source): You can put this solution on YOUR website!I see at least three different ways to read this. Is the question 1)"How many ways can a student answer each question?" meaning say only question 6 4 ways per question or 2)"How many ways can a student answer all of the questions?" a) 1 way because he can only hand in one test with one set of answers b) 5 because he can only hand in one test with one set of answers but he must choose 5 different answers or c)20 = 5*4 those are the choices before the student. or 3) "How many different ways can the test be answered?" 4^5= 4*4*4*4*4=1024 There are 4 choices for 5 questions. And each choice is independent of the others.
 Travel_Word_Problems/259533: Which rate is faster: a bird traveling at 2 kilometers per hour, or an insect traveling at 1,000 meters per hour?1 solutions Answer 191682 by richwmiller(9135)   on 2010-01-24 09:43:53 (Show Source): You can put this solution on YOUR website!1 kilometer =1000 meters
 Equations/260246: please solve 2/(t+6)=3/(t-1)1 solutions Answer 191681 by richwmiller(9135)   on 2010-01-24 09:41:52 (Show Source): You can put this solution on YOUR website!If you mean multiply by t 2+6t=3-t 7t=1 t=1/7 If you mean 2(t-1) = 3(t+6) 2t-2=3t+18 add 2 and subtract 2t t=-20
 Geometry_Word_Problems/260243: Fencing dog pens. Clint is constructing two adjacent rectangular dog pens. Each pen will be three times as long as it is wide, and the pens will share a common long side. If Clint has 65 ft of fencing, what are the dimensions of each pen?1 solutions Answer 191676 by richwmiller(9135)   on 2010-01-24 09:33:50 (Show Source): You can put this solution on YOUR website!3W=L 3L+4W=65 L=15 W=5 3*15+4*5=65 45+20=65
 Mixture_Word_Problems/260242: Alcohol solutions. Gus has on hand a 5% alcohol solution and a 20% alcohol solution. He needs 30 liters of a 10% alcohol solution. How many liters of each solution should he mix together to obtain the 30 liters?1 solutions Answer 191674 by richwmiller(9135)   on 2010-01-24 09:28:06 (Show Source): You can put this solution on YOUR website!.05x+.2(30-x)=30*.1 20 liters of 5% 10 liters of 20%
 Travel_Word_Problems/260245: Two buses leave Wilmington at the same time and travel in opposite directions. One bus averages 55mph and the other bus averages 45mph. When will they be 400 miles apart?1 solutions Answer 191673 by richwmiller(9135)   on 2010-01-24 09:25:18 (Show Source): You can put this solution on YOUR website!Wilmington isn't big enough for two busses at the same time! 55t+45t=400 100t=400 t=4 hours later
 Equations/260244: x^2-3x=01 solutions Answer 191672 by richwmiller(9135)   on 2010-01-24 09:22:50 (Show Source): You can put this solution on YOUR website!x(x-3)=0 x=0 x=3
 Percentage-and-ratio-word-problems/260216: In a school, the ratio of boys to girls is 3 : 5. After 20 more boys attended the school, the ratio is now 1 : 1. How many girls are in the school?1 solutions Answer 191671 by richwmiller(9135)   on 2010-01-24 09:21:13 (Show Source): You can put this solution on YOUR website!3/5=b/g b+20=g b=30 g=50
 Miscellaneous_Word_Problems/260214: In a field of 108 aliens, each alien has either 5 heads or 103 heads. If I counted 1030 heads in total, how many aliens have 103 heads?1 solutions Answer 191669 by richwmiller(9135)   on 2010-01-24 09:12:01 (Show Source): You can put this solution on YOUR website!a+b=108 5a+103b=1030 a=103 and b=5 check 5*103 +103*5=1030 515+515=1030 ok
 Geometry_Word_Problems/260219: In a circle with radius 13, two parallel chords with length 10 are joined to create a rectangle. Find the area of this rectangle. 1 solutions Answer 191667 by richwmiller(9135)   on 2010-01-24 09:08:16 (Show Source): You can put this solution on YOUR website!one side is 10 the diagonal is 26 26^2=10^2+b^2 676=100+b^2 576=b^2 24=b 10*24=240 square units is the area
Equations/260240: 4x^2-49=0
1 solutions

Answer 191665 by richwmiller(9135)   on 2010-01-24 09:00:06 (Show Source):
You can put this solution on YOUR website!
find factors
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 4 and the last term -49 to get -196. Now we need to ask ourselves: What two numbers multiply to -196 and add to 0? Lets find out by listing all of the possible factors of -196

Factors:

1,2,4,7,14,28,49,98,196,

-1,-2,-4,-7,-14,-28,-49,-98,-196, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -196.

(-1)*(196)=-196

(-2)*(98)=-196

(-4)*(49)=-196

(-7)*(28)=-196

(-14)*(14)=-196

Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0

||||||||||
 First Number | Second Number | Sum 1 | -196 | 1+(-196)=-195 2 | -98 | 2+(-98)=-96 4 | -49 | 4+(-49)=-45 7 | -28 | 7+(-28)=-21 14 | -14 | 14+(-14)=0 -1 | 196 | (-1)+196=195 -2 | 98 | (-2)+98=96 -4 | 49 | (-4)+49=45 -7 | 28 | (-7)+28=21 -14 | 14 | (-14)+14=0

We can see from the table that 14 and -14 add to 0. So the two numbers that multiply to -196 and add to 0 are: 14 and -14

breaks down to this (just replace with the two numbers that multiply to -196 and add to 0, which are: 14 and -14)

Replace with

Group the first two terms together and the last two terms together like this:

Factor a 2x out of the first group and factor a -7 out of the second group.

Now since we have a common term we can combine the two terms.

Combine like terms.
==============================================================================

Notice how foils back to our original problem . This verifies our answer.

 Rate-of-work-word-problems/260221: A crew of 30 people can build a certain road in 60 days. After the tenth day the plans are changed; the company wants the road built in 30, not 60 days. How many more people must be hired? 1 solutions Answer 191662 by richwmiller(9135)   on 2010-01-24 08:57:02 (Show Source): You can put this solution on YOUR website!30*60=1800 man days 30*10=300 man days expended 1800-300=1500 man days to go 1500/20=75 workers 75-30=45 more workers to be hired
3x^2-7x+3=0

1 solutions

Answer 191657 by richwmiller(9135)   on 2010-01-24 08:37:38 (Show Source):
You can put this solution on YOUR website!
 Solved by pluggable solver: SOLVE quadratic equation with variable Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=13 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.767591879244, 0.565741454089335. Here's your graph:

 Mixture_Word_Problems/260227: 300 gms of sugar solution contains 40% sugar. How much sugar should be added to make it 50% sugar solution?1 solutions Answer 191656 by richwmiller(9135)   on 2010-01-24 08:37:34 (Show Source): You can put this solution on YOUR website!300*.4+x=(300+x)*.5 x=60 60 grams
 Problems-with-consecutive-odd-even-integers/260234: The sum of five consecutive integers is 40. What is the smallest of the integers?1 solutions Answer 191651 by richwmiller(9135)   on 2010-01-24 08:29:52 (Show Source): You can put this solution on YOUR website!n+n+1+n+2+n+3+n+4=40 5n+10=40 5n=30 n=6 the smallest is 6 6,7,8,9,10
1 solutions

Answer 191649 by richwmiller(9135)   on 2010-01-24 08:25:34 (Show Source):
You can put this solution on YOUR website!
First thing is to see if it can be factored.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 1 and the last term -60 to get -60. Now we need to ask ourselves: What two numbers multiply to -60 and add to 11? Lets find out by listing all of the possible factors of -60

Factors:

1,2,3,4,5,6,10,12,15,20,30,60,

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -60.

(-1)*(60)=-60

(-2)*(30)=-60

(-3)*(20)=-60

(-4)*(15)=-60

(-5)*(12)=-60

(-6)*(10)=-60

Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11

||||||||||||
 First Number | Second Number | Sum 1 | -60 | 1+(-60)=-59 2 | -30 | 2+(-30)=-28 3 | -20 | 3+(-20)=-17 4 | -15 | 4+(-15)=-11 5 | -12 | 5+(-12)=-7 6 | -10 | 6+(-10)=-4 -1 | 60 | (-1)+60=59 -2 | 30 | (-2)+30=28 -3 | 20 | (-3)+20=17 -4 | 15 | (-4)+15=11 -5 | 12 | (-5)+12=7 -6 | 10 | (-6)+10=4

We can see from the table that -4 and 15 add to 11. So the two numbers that multiply to -60 and add to 11 are: -4 and 15

breaks down to this (just replace with the two numbers that multiply to -60 and add to 11, which are: -4 and 15)

Replace with

Group the first two terms together and the last two terms together like this:

Factor a 1x out of the first group and factor a 15 out of the second group.

Now since we have a common term we can combine the two terms.

Combine like terms.
==============================================================================