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# Recent problems solved by 'venugopalramana'

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 Trigonometry-basics/20853: Please let me know if I'm on the right track with this. A plane is observed approaching your home and you assume the its speed is 550 km/hr. The angle of elevation of the plane is 16 degrees at one time and 57 degrees one minute later. Approximate the altitude of the plane. Let r be the approximate altitude of the plane. 41 degrees = 9.17 km 360 degrees = 80.52 km 80.52=2pier r=80.52/2pie r=12.82 km Thank you!1 solutions Answer 10007 by venugopalramana(3286)   on 2005-11-29 00:17:50 (Show Source): You can put this solution on YOUR website!SEE MY COMMENTS IN CAPITAL LETTERS A plane is observed approaching your home and you assume the its speed is 550 km/hr. ......OK..... The angle of elevation of the plane is 16 degrees at one time......OK and 57 degrees .....OK one minute later....OK Approximate the altitude of the plane....OK Let r be the approximate altitude of the plane.....OK I AM NOT ABLE TO FOLLOW THE LATER PART AS YOU HAVE NOT GIVEN A SKETCH..BUT IT LOOKS AS IF YOU ASSUMED A CIRCULAR FIGURE OR SO....OK ..LET ME TRY...LET YOUR POSITION ON GROUND BE A.LET THE PLANE BE AT P FIRST AND Q LATER.SO PQ = DISTANCE TRAVELLED BY PLANE IN 1 MINUTE=550/60=55/6 KM.FROM P AND Q DROP PERPENDICULARS PB AND QC TO GROUND LEVEL WHERE YOU ARE.SO PB=QC=ALTITUDE OF PLANE = R ASSUMED. APB IS A RIGHT ANGLED TRIANGLE WITH ANGLE PAB=16...HENCE TAN 16 =R/AB.... AB=R/TAN(16) SIMILARLY AC=R/TAN(57) BC=AB-AC=R/TAN(16)-R/(TAN(57)=R(1/TAN16-1/TAN57)..BUT...BC=PQ=55/6..SO WE HAVE 55/6=R(1/TAN16-1/TAN57)...FROM TABLES YOU CAN FIND TAN 16 AND TAN 57 AND USING THOSE VALUES YOU CAN FIND R FROM THE ABOVE EQUATION.
 Trigonometry-basics/20855: Please let me know if I'm on the right track with this. A plane is observed approaching your home and you assume the its speed is 550 km/hr. The angle of elevation of the plane is 16 degrees at one time and 57 degrees one minute later. Approximate the altitude of the plane. Let r be the approximate altitude of the plane. 41 degrees = 9.17 km 360 degrees = 80.52 km 80.52=2pier r=80.52/2pie r=12.82 km Thank you!1 solutions Answer 10006 by venugopalramana(3286)   on 2005-11-29 00:14:12 (Show Source): You can put this solution on YOUR website!SEE MY COMMENTS IN CAPITAL LETTERS A plane is observed approaching your home and you assume the its speed is 550 km/hr. ......OK..... The angle of elevation of the plane is 16 degrees at one time......OK and 57 degrees .....OK one minute later....OK Approximate the altitude of the plane....OK Let r be the approximate altitude of the plane.....OK I AM NOT ABLE TO FOLLOW THE LATER PART AS YOU HAVE NOT GIVEN A SKETCH..BUT IT LOOKS AS IF YOU ASSUMED A CIRCULAR FIGURE OR SO....OK ..LET ME TRY...LET YOUR POSITION ON GROUND BE A.LET THE PLANE BE AT P FIRST AND Q LATER.SO PQ = DISTANCE TRAVELLED BY PLANE IN 1 MINUTE=550/60=55/6 KM.FROM P AND Q DROP PERPENDICULARS PB AND QC TO GROUND LEVEL WHERE YOU ARE.SO PB=QC=ALTITUDE OF PLANE = R ASSUMED. APB IS A RIGHT ANGLED TRIANGLE WITH ANGLE PAB=16...HENCE TAN 16 =R/AB.... AB=R/TAN(16) SIMILARLY AC=R/TAN(57) BC=AB-AC=R/TAN(16)-R/(TAN(57)=R(1/TAN16-1/TAN57)..BUT...BC=PQ=55/6..SO WE HAVE 55/6=R(1/TAN16-1/TAN57)...FROM TABLES YOU CAN FIND TAN 16 AND TAN 57 AND USING THOSE VALUES YOU CAN FIND R FROM THE ABOVE EQN. 41 degrees = 9.17 km 360 degrees = 80.52 km 80.52=2pier r=80.52/2pie r=12.82 km
 Triangles/20859: Given the coordinates of the midpoints of the sides of a triangle find the coordinates of the vertices of the triangle. Midpoint Coordinates: L(-2,1) M(2,3) N(3,-2) I know that you can plot the coordinates and then draw in the triangle and find the coordinates of the vertices that way, but I was wondering if there was an easier or mathematical way to go about solving the problem. Thank you for looking this over.1 solutions Answer 10004 by venugopalramana(3286)   on 2005-11-28 23:48:59 (Show Source): You can put this solution on YOUR website!Midpoint Coordinates: L(-2,1) M(2,3) N(3,-2) LET ABC BE THE TRIANGLE WITH COORDINATES OF VERTICES AS A(X1,Y1),B(X2,Y2),C(X3,Y3).LET L,M,N BE THE MIDPOINTS OF AB,BC,CA RESPECTIVELY.AS PER COORDINATE GEOMETRY MIDPOINT'S COORDINATES ARE GIVEN BY THE ARITHMATIC MEAN OF COORINATES OF THE END POINTS..SO.. L={(X1+X2)/2,(Y1+Y2)/2} HENCE (X1+X2)/2=-2...OR X1+X2=-2*2=-4 X1+X2=-4......(1) SIMILARLY WE GET FROM M AND N X2+X3=4.........(2) X3+X1=6...........(3) ADDING ALL 3 EQNS.1,2,3,WE GET X1+X2+X2+X3+X3+X1=2(X1+X2+X3)=-4+4+6=6 X1+X2+X3=3.......(4) NOW SUBTRACTING EQNS..1,2,AND 3 FROM THIS ,WE GET X3=7 X2=-1 X1=-3 SIMILARLY YOU CAN FIND Y1,Y2,Y3 WHICH WILL COMPLETE YOUR REQUIREMENT..
 Complex_Numbers/20861: can u help me with this specific problem i dont understand it its 4(7-i)-5(2-6i) i would greatly appreciate it thanks1 solutions Answer 10001 by venugopalramana(3286)   on 2005-11-28 23:25:43 (Show Source): You can put this solution on YOUR website! 4(7-i)-5(2-6i) HOPE YOU KNOW COMPLEX NUMBERS...FOR ELEMENTARY UNDERSTANDING IT SUFFICES TO KNOW THAT WE DESIGNATE SQUARE ROOT OF -1 AS i WHICH IS AN IMAGINARY NUMBER..WHERE AS 1,2,3 ETC..ARE ALL REAL NUMBERS..WE CANT MIX THEM LIKE WE CANT ADD 2 APPLES TO 3 GRAPES AND SAY THAT THERE ARE 5 APPLES OR 5 GRAPES..WE CAN ONLY SAY THAT THERE ARE 2 APPLES AND 3 GRAPES..SO WE DEAL WITH REAL NUMBERS AND IMAGINARY NUMBERS SEPERATELY AND CALL THEM TOGETHER AS COMPLEX NUMBERS 4(7-i)-5(2-6i)=4*7-4i-(5*2-5*6i)=28-4i-10+30i=(28-10)+i(30-4)=18+26i
 Linear-equations/20870: I'm stumped. Solve for Y 4(2y-5)=2(4y-10) 8y-20=8y-20 8y-20+20=8y-20+20 8y=8y 1/8*8y=1/8*8y y=y Were am I going wrong?1 solutions Answer 9999 by venugopalramana(3286)   on 2005-11-28 23:16:31 (Show Source): You can put this solution on YOUR website!THERE IS NOTHING WRONG WITH YOUR ANSWER..THE PROBLEM GIVEN IS LIKE THAT.. IT IS AN IDENTITY...WHEN LEFT HAND SIDE OF AN EQUATION IS SAME AS THAT ON THE RIGHT HAND SIDE IT IS CALLED AN IDENTITY..IT IS TRUE FOR ANY VALUE OF THE VARIABLE INVOLVED.SO YOU CAN SAY THAT Y = 0,1,-1,1/2....ETC..ANY AND EVERY VALUE. SOME MORE EXAMPLES OF IDENTITIES ARE (X+Y)^2=X^2+Y^2+2*X*Y WHICH IS TRUE FOR ANY VALUE OF X AND Y.
 logarithm/20718: CAN SOMEONE, ANYONE PLEASE PLEASE HELP ME!!! I really need help. What is the value of X in the equation, log(2x+8) - log(2x^2+21x+61) = -31 solutions Answer 9980 by venugopalramana(3286)   on 2005-11-28 11:41:58 (Show Source): You can put this solution on YOUR website!i think you mean logs are to base 2..taking that and using the formula, log x to base y = log x/log y so we get log(2x+8)/log 2+log (2x^2+21x+61)/log 2=-3 multiplying through out with log 2 log(2x+8)+log(2x^2+21x+61)=-3log 2=log 2^(-3)==log(1/2^3)=log(1/8) log(2x+8)(2x^2+21x+61)=log(1/8) (2x+8)(2x^2+21x+61)=(1/8) i hope now you can continue from here to simplify and factorise to get the answers
 Permutations/20719: Five years ago, \$10000 was invested at 6%/a compounded semi-annually. Today the investment rates have risen to 7%/a compounded annually. If the original investment and accumulated interest is rolled into the new investment conditions, how much will it be worth in five years. Use A=P(1+i)^n I know for 6% P=10000, i = 0.03 but I don't know what n =? I know for 7% P=10000, i = 0.07 But I don't know what n =?1 solutions Answer 9978 by venugopalramana(3286)   on 2005-11-28 11:31:29 (Show Source): You can put this solution on YOUR website!SEE MY COMENTS IN CAPITAL LETTERS ............................................................................. > Five years ago, \$10000 was invested at 6%/a > compounded semi-annually. Today > the investment rates have risen to 7%/a compounded > annually. If the original > investment and accumulated interest is rolled into > the new investment > conditions, how much will it be worth in five years. > > Use A=P(1+i)^n..HERE A=AMOUNT=PRINCIPAL+INTEREST..P=PRINCIPAL YOU USED I FOR RATE PER TERM PER DOLLAR.. WE USE R FOR RATE AND I FOR INTEREST..N=NUMBER OF TERMS... > I know for 6% P=10000,CORRECT i = 0.03 VERY GOOD..IT IS RATE PER TERM HERE TERM IS SEMIANNUAL SO INTERET RATE IS HALF OF 6% THAT IS 3% THAT IS = 0.03 PER DOLLAR..but I don't know > what n =? AS I SAID N = NUMBER OF TERMS FOR 5 YEARS =10 HALF YEARS....SO N=10 > I know for 7% P=10000,NO...IT IS ACUMULATED INTEREST AND PRINCIPAL..SO YOU SHOULD USE A OBTAINED FROM ABOVE CALCULATION AS P HERE. i = 0.07 VERY GOOD..But I don't know > what n =?..N=5 IN THIS CASE AS NUMBER OF TERMS IN 5 YEARS IS 5. > NOW FIND A USING THE ABOVE VALUES.
 logarithm/20724: I REALLY REALLY NEED HELP A.S.A.P. What is the value of N in the equation: log(n-3) + log(n+4) = 1. I REALLY REALLY appreciate the help because I really need it. Sorry for the short notice.1 solutions Answer 9977 by venugopalramana(3286)   on 2005-11-28 11:27:19 (Show Source): You can put this solution on YOUR website!i think you mean logs to base 8 ...assuming that and using the formula log x to base y = log x /log y...where all logs are taken to any common base. so we have log(n-3)/log 8 + log(n+4)/log 8 =1....multipling through out with log 8,we get log (n-3)+log (n+4)=log 8 log(n-3)(n+4)=log 8 hence (n-3)(n+4)=8 n^2-3n+4n-12-8=0 n^2+n-20=0 n^2+5n-4n-20=0 n(n+5)-4(n+5)=0 (n+5)(n-4)=0 hence n=-5 or n=4
 logarithm/20664: I am getting really frustrated because I don't know how to go about solving this at all. Use log5 = 0.6990 and log7 = 0.8451 to evaluate log0.05 PLEASE walk me through this!!1 solutions Answer 9942 by venugopalramana(3286)   on 2005-11-27 11:03:54 (Show Source): You can put this solution on YOUR website!you mean log 5 to base 10=0.6990 etc..since all logs are to base 10 we may take it for granted and drop writing that base..so we have log 5 = 0.6990 log 7 =0.8451 to find log 0.05 log 0.05=log (5/100)=log 5 - log 100 = 0.6990 - 2 = -1.3010... since log 100 to base 10 =2...note that 10^2 = 100 .... hence by definition log 100 to base 10 =2..
 logarithm/20635: 2(3)^x=5^(x-1) I dont understand how to find x , can someone please help me?1 solutions Answer 9940 by venugopalramana(3286)   on 2005-11-27 10:44:38 (Show Source): You can put this solution on YOUR website!2(3)^x=5^(x-1) trust you know logarithms...take logs on both sides ...use the formulae.. log x*y = log x + log y log x^n= n*log x taking logs we get log 2+ log 3^x=(x-1)*log 5 log2+x log 3=(x-1) log 5=x log 5 - log 5 x(log 5-log 3)=log 2+ log 5=log 2*5 = log 10 =1 x log(5/3)=1 x = 1/log(5/3)
 logarithm/20603: Please help me with this problem: logb^2 = .25596, logb^3 = .40568, logb^5 = .59432 logb^(5/12)^3 I know the answer is .096984, but don't know how to go about it. Thanks, Sandy1 solutions Answer 9920 by venugopalramana(3286)   on 2005-11-26 01:29:28 (Show Source): You can put this solution on YOUR website!logb^2 = .25596, logb^3 = .40568, logb^5 = .59432 this is not properly written..i think by the above you meant log 2 to base b =0.25596..etc....then the above has some meaning..otherwise they are not correct and represent contradictory things logb^(5/12)^3=?...as given above this should mean that we want log (5/12)^3 to base b. use the following formulae in logs which are true for any base..here wew can take b as base log x*y=log x+log y log x/y=log x-log y log x^n=n*log x log x to base b=log x /log b LET US TAKE ALL LOGS TO BASE b AS WE HAVE THEM TO BASE b IN THE PROBLEM. log(5/12)^3=3*log(5/12) =3*(log 5-log 12)=3*(log 5-log 2*2*3) =3*(log 5-log 2-log 2-log 3)=3*(0.59432-0.25596-0.25596-0.40568) 3*(-0.3238)=-0.96984 YOU GAVE THE ANSWER AS 0.096984...IN WHICH CASE THE QUESTION SHOULD BE log(5/12)^(-0.3) to base b =????.PLEASE CHECK BACK THE QUESTION WRITE IT DOWN PROPERLY AS AS GIVEN IN THE TEXT AND COME BACK
 Matrices-and-determiminant/20522: How do I perform the next required row operation on the following matrix and provide only the next table: x y z 1 28 14 245 0 3 7 42 0 7 7 -381 solutions Answer 9892 by venugopalramana(3286)   on 2005-11-25 08:01:32 (Show Source): You can put this solution on YOUR website!trust you want to solve the equations for x,y and z and you are at this stage now....assuming that .....our objective is to finally get the matrix if possible into the following form ....(i am using ....to seperate the numbers with suitable gaps..your typing is giving raise to uneven gaps bringing a little lack of clarity) 1.....0.....0.....x 0.....1.....0.....y 0.....0.....1.....z now we have 1......28.....14.....245 0.......3......7......42 0.......7......7.....-38 new row2=old row2/3.......to get 1 as required in row2.so we get... 1......28.....14.....245 0......3/3....7/3....42/3 0.......7......7.....-38 new row3=oldrow3-7*row2 to get 0 as required in row3 1......28.....14...........245 0.......1.....7/3...........14 0......7-7*1..7-7*7/3......-38-7*14 new row3 = old row3/(-28/3)..to get 1 as required in row3 1......28.....14...................245 0.......1.....7/3...................14 0.......0....(-28/3)/(-28/3).....(-136)/(-28/3) this gives us finally in the following form 1......28.....14............245 0.......1.....7/3...........14 0.......0......1............102/7 now we go back in the same way to get 0 in row2 and row3 new row2=old row2-row3*7/3...and new row1=old row1-row3*14...so we get 1......28......14-1*14.......245-(102/7)*14 0.......1.......7/3-(7/3)*1...14-(102/7)*(7/3) 0.......0.........1.............102/7 the above on simplification gives us 1.......28.......0..........41 0........1.......0..........-20 0........0.......1..........102/7 now finally we take new row1=old row1-28*row2 1.......28-28*1......0.......41-(-28*20) 0........1.......0...........-20 0........0.......1...........102/7 so the final answer is 1......0.......0.......601 0......1.......0.......-20 0......0.......1.......102/7 which tells us that 1*x+0*y+0*z=x=601 0*x+1*y+0*z=y=-20 0*x+0*y+1*z=z=102/7 note that each and every transformation we did above can be interpreted as given in the last statement given above...this i hope will give you the insight of the process at every step.you can also substitute these values of x,y and z in each and every matrix above to see that they satify all the equations given by the different matrices..in general each mtrix can be taken as a set of simltanous equations in x,y and z...they can be written as follows..take column 1 is for x,column 2 is for y and column 3 is for z.so the first matrix you gave 1......28.....14.....245 0.......3......7......42 0.......7......7.....-38 tells us that 1*x+28*y+14*z=245....etc...
 Numbers_Word_Problems/20527: Claire has a small personal business making scented candles. Her startup costs are \$200. Her cost to make each cangle is \$1.50. She sells each candle for \$4.00. Use a system of equations to determine the number of candles she must sell to break even. (Write one equation for her prodction costs and one for her sales revenue.)1 solutions Answer 9891 by venugopalramana(3286)   on 2005-11-25 06:24:45 (Show Source): You can put this solution on YOUR website!production cost...all figures in \$.... basis...number of candles made = x start up cost =200 making cost per candle =1.5 cost of making x candles =1.5*x total production cost =200+1.5*x sales revenue.... sale price per candle=4 sales revenue for x candles =4*x now to break even............. sales revenue =production cost so.........4*x=200+1.5*x 4*x-1.5*x=200 2.5*x=200 x=200/2.5=80 hence 80 candles are to be made to break even..
 Numbers_Word_Problems/20528: Sales clerk at an electronics store have a choice of two methods of pay: Plan A: weekly salary of \$450 plus 3% commission on sales Plan B: a straight 7% commission on sales Write a system of equations for each plan.1 solutions Answer 9890 by venugopalramana(3286)   on 2005-11-25 05:56:48 (Show Source): You can put this solution on YOUR website!Basis...1 week...let sales be \$ x plan A.... 3% commission on sales =3*x/100 weekly salary = \$ 450 total emoluments in \$ =450+3*x/100 plan B....... 7% commission on sales =7*x/100 total emoluments in \$=7*x/100
 Expressions-with-variables/20538: I need to solve the following: Write 8 expressions that have different results when they are evaluated. You must use all three numbers in each expression and each number can only be used once. Thanks, Teri1 solutions Answer 9889 by venugopalramana(3286)   on 2005-11-25 05:36:13 (Show Source): You can put this solution on YOUR website!the problem is not very explicit..what are the 3 numbers ?..let us take them as 2,3,4.so let us write 8 different expressions using these 3 numbers using each number once only in each expression which give different answers... 1.....2+3+4=9 2.....2+3-4=1 3........2-3-4=-5 4.........2*3*4=24 5........2*(3+4)=14 6...........2*(3-4)=-2 7..........2*(4-3)=2 8.......3*4/2=6
 Equations/20536: I need to construct a set of twelve numbers with a mean of 7 a median of 6 and a mode of 8. Can you help. Teri1 solutions Answer 9888 by venugopalramana(3286)   on 2005-11-25 05:20:38 (Show Source): You can put this solution on YOUR website!there are 12 numbers..say x1,x2,x3,.....x11,x12 when arranged in increasing order. mean is 7.so sum of numbers by /total no.of numbers=7=(x1+x2+....+x11+x12)/12 hence sum of numbers =7*12=84=(x1+x2+....+x11+x12) median is 6..that is the middle term is 6..but total number of terms is 12 so there would be 2 middle terms...namely x6 and x7...by convention in such cases their average is taken as median..so (x6+x7)/2=6...or....x6+x7=2*6=12. mode is 8 ...that is the frequency of occurence of mode or number 8 should be maximum... now there are no unique answers for this... we are just to give any set of 12 numbers satisfying the above 3 conditions. say one set of numbers could be then .... let us put median as 6 ..by taking both x6 and x7 as 6 each... now 8 should occur maximm numer of times ...so let x8=x9=x10=x11=8..here 8 occurs 4 times. let us take x12=16 say... total of numbers we have taken so far is 6+6+8*4+16=60 now take x1 to x5 so that thy add up to 84-60=24...since total should be 84 to get mean 7 as given above...we only have to see that no number should occur more than 4 times as mod of 8 ,we took with frequency of 4 which should be the highest... so let us take x1=4,x2=4,x3=5,x4=5,x5=6..which satisfies all the above conditions so the set of numbers in increasing order are 4,4,5,5,6,6,6,8,8,8,8,16.... their mean is (4+4+5+5+6+6+6+8+8+8+8+16)/12=84/12=7 median is 6 ,since middle most number /numers are 6. mode is 8 as 8 occurs 4 times ,the maximum frequency..
 Exponential-and-logarithmic-functions/20113: i have a logarithmic graph ......... how do i find the equation?1 solutions Answer 9710 by venugopalramana(3286)   on 2005-11-19 01:16:25 (Show Source): You can put this solution on YOUR website!do you mean that y axis is plotted on log scale and x axis is plotted on normal scale..or both x and y axis are plotted to log scale ,the folowing example will help to understand let us take an example..let the graph be a straight line while plotted on log scale for y axis and normal scale for x axis set of data could be x.....1....10.....100......1000......10000....etc log x.....0.....1.......2........3..........4....etc... y.....10....1.......1/10.....1/100.......1/1000...etc... log y..1....0........-1.......-2............-3...... while taking a scale of 1 cm as 1 unit you will find that in log graph ,if there is 1 at 0 cm,the next point after 1 cm will show 10 , and the next point after another cm will show 100 ,the next point after another 1 cm will show 1000 , and the next point after another cm will show 10000...etc..that is ineffect they are log 1 at zero cm,log 10 at 1 cm ,log 100 at 2 cm ...etc... hence if we get a straight line on such a plot ...that means really ,assuming both axes are plotted to lg scale,log x is linearly related to log y...or log y =m*log x+log c...where m is the slope of the line and log c is the y intercept of the line ..drawing comparison from linear scale plots wher we call the eaqn.as y=mx+c so on a log plot instead of y=mx+c we have log y = m log x +log c that is using log formula...log x*y=log x +log y,and a log x = log (x^a)we get log y =log (x^m)*c y=(x^m)*c...is the real eqn.without logs ..so the key is take the log given graph ...see its pattern and make the eqn in normal manner and finally replace x with log x and y with log y to get the real relation ship..simplify if neede as above to get eqn. without logs. hope it is clear if you have any difficulty please come back ...
 Complex_Numbers/20119: how do you find out inmaginary numbers 1 solutions Answer 9709 by venugopalramana(3286)   on 2005-11-19 00:49:03 (Show Source): You can put this solution on YOUR website!SEE THE FOLLOWING EXAMPLE TO KNOW Is the square root of 9 equal to -3i? 1 solutions -------------------------------------------------------------------------------- Answer 9707 by venugopalramana(356) on 2005-11-19 00:41:47 (Show Source): NO NO >>> -3i*-3i=+9*(i*i)=9*(-1)=-9 square root of a negative number only gives raise to complex numbers involving i..square root of a positive number is always real and does not involve complex or imaginary number i so square root of +9 could be +3 or -3 where as square root of -9 is +3i or -3i...i being defined with the property i*i=-1...which as you can see from above is an imaginary number
 Graphs/20114: Help please What is the domain of f in Is it the set of Real numbers?1 solutions Answer 9708 by venugopalramana(3286)   on 2005-11-19 00:45:49 (Show Source): You can put this solution on YOUR website!yes....you are correct ..there is no limitation on x for the function e^(x+1)to be meaning full or legal mathematically so set of real numbers is the domain
 Complex_Numbers/20120: Is the square root of 9 equal to -3i? 1 solutions Answer 9707 by venugopalramana(3286)   on 2005-11-19 00:41:47 (Show Source): You can put this solution on YOUR website!NO NO >>> -3i*-3i=+9*(i*i)=9*(-1)=-9 square root of a negative number only gives raise to complex numbers involving i..square root of a positrive number is always real and does not involve complex or imaginary number i so square root of +9 could be +3 or -3
 Complex_Numbers/20129: Linear systems: 1 solutions Answer 9706 by venugopalramana(3286)   on 2005-11-19 00:36:17 (Show Source): You can put this solution on YOUR website!SEE THE FOLLOWING EXAMPLE AND DO THE SAME WAY>>> 2x-3y=5 4x-6y=10 the second eqn. is obtainedby multiplying with 2 the first eqn. so they are dependent eqns..so in effect we have only one eqn.with 2 variables which give raise to infinte solutions let us take 2x-3y=5 put x=0,1,2,3,etc..and get corresponding values of y x..0......1......2...3...4...5.. y..-5/3...-1...-1/3.....etc..infinite solutions .
 logarithm/20128: 5^(log base5 15)1 solutions Answer 9705 by venugopalramana(3286)   on 2005-11-19 00:33:48 (Show Source): You can put this solution on YOUR website!let 5^(log base5 15)=x...so taking logs to a common base 10 say,we get [using the formula log x to base y =log x / log y ] {log base5 (15)}log 5 =logx (log15/log5)log 5 = logx log 15 = log x so x=15
 Complex_Numbers/20127: 1 solutions Answer 9704 by venugopalramana(3286)   on 2005-11-19 00:25:04 (Show Source): You can put this solution on YOUR website! the second eqn. is obtainedby multiplying with 2 the first eqn. so they are dependent eqns..so in effect we have only one eqn.with 2 variables which give raise to infinte solutions let us take 2x-3y=5 put x=0,1,2,3,etc..and get corresponding values of y x..0......1......2...3...4...5.. y..-5/3...-1...-1/3.....etc..infinite solutions .