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in echelon form, tell how many solutions there are in nonnegative integers. x+3y+z=82 7y+2z=42

For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers x+3y+z=82 7y+2z=42

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... For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers. x+3y+z=82. 7y+2z=42. 1b) ... 7x-3y+z=8. 5y-9z=-4 ...
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you please help me with some math problems from ... equations in echelon form, tell how many solutions there are in nonnegative integers. x+3y+z=82. 7y+2z=42. 1b) ...
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Tutorials Hub. Menu. ... For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers. x+3y+z=82. 7y+2z=42. 1b) ...
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Tutorials Hub. Menu. ... For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers. x+3y+z=82. 7y+2z=42. 1b) ...
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Find the reduced row-echelon form of the augmented matrix that represents the ... {x + 2y + z = -17 {3x + 7y + 2z = -56 {x - y ... Round your answer to the nearest ...
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... in which case, there are infinitely many solutions. ... reduced row-echelon form. Then there are ... 2 Since x, y, and z must all be integers ...
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5 Systems and Matrices. We found 20 results related to this asset. Document Information; Type: eBook; Total # of pages: 112. Avg Rating: Textbook Information.
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in echelon form, tell how many solutions there are in nonnegative integers. x+3y+z=82 7y+2z=42

????????? "echelon form" ???????? ??? yu hav NE idea wot "echelon" meen ????
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Tell wether the system has one solution, infinetly many solutions or no solutions. 2x-5y=17 and 6x-15y=15

Tell whether the system has one solution, infinitely many solutions or no solutions. 2x-5y=17 and 6x-15y=51 The equations are, 2x – 5y = 17 6x – 15y = 51 Putting them into standard form, y = (2/5)x – 17/5 y = (6/15)x – 51/15 = (2/5)x – 17/5 The two equations are equations of straight lines with the same slope (m = 2/5) and with the same y-intercept (c = -17/5). This means that the two lines are co-linear. They lie on top of each other, so there are an infinite number of points of intersection. Answer: There are infinitely many solutions.
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Shawn took in $69.15 in two hours work on Saturday selling Belts for $8.05 and earrings for $4.50. How many of each did Shawn sell?

Shawn took in $69.15 in two hours work on Saturday selling Belts for $8.05 and earrings for $4.50. How many of each did Shawn sell? I'm not too sure how ​you are expected to solve this problem. You will end up with a Diophantine Equation (one involving integer coefficients and integer solutions only) If you have never heard of Diophantus, then the precise mathematical solution later on should be ignored. Instead ...   The (Diophatine) equation is 161B + 90E = 1363 You should be able to show that Shawn can only sell a maximum of 8 belts (i.e. if he sells no earrings), and a maximum of 15 earrings (i.e. if he sells no belts). So start with the smaller number (8). Shawn sells up to 8 belts. So put B = 1 in the (Diophantine ) equation and see if B is an integer. If not, then B=1 cannot be a solution, So now try again with B = 2, and so on ..     My precise mathematical solution now follows. Let B be the number of belts sold @ $8.05 per belt. Let E be the number of earrings sold @ $4.50 per pair. Total money made is P = $69.15 Total money made is: B*$8.05 + E*$4.50 Equating the two expressions, 8.05B + 4.50E = 69.15 Making the coefficients as integers we end up with a Diophantine equation. 161B + 90E = 1383 We now manipulate the coefficients 161 and 90. We should end up with a relation between the two that should produce the value 1. This manipulation is in two parts. In the first line, we are writing the larger coefft, 161, as a multiple of the smaller coefft plus a remainder. 1st Part 161 = 1x90 + 71  90 = 1x71 + 19     now we write the 1st multiple as a multiple of the 1st remainder + remainder 71 = 3x19 + 14    we write the 2nd multiple as a multiple of the 2nd remainder + remainder 19 = 1x14 + 5 14 = 2x5 + 4 5 = 1x4 + 1 ============ 2nd Part We rewrite that last equation so that 1 is on the left hand side 1 = 5 – 1x4 We now use the remainder from the next equation up. 1 = 5 – 1x(14 – 2x5) 1 = 3x5 – 1x14 And again we use the remainder in the next equation up 1 = 3(19 – 1x14) – 1x14 1 = 3x19 – 4x14 1 = 3x19 – 4(71 – 3x19) 1 = 15x19 – 4x71 1 = 15(90 – 1x71) – 4x71 1 = 15x90 – 19x71 1 = 15x90 – 19(161 – 1x90) 1 = 34x90 – 19x161 ================ Therefore, 1383 = (-19x1383).161 + (34x1383).90 1383 = -26,277*161 + 57,022*90 Which implies B = -26,277 E = 47,022 which is obviously incorrect! But all is not lost. These two values for B and E are simply two values that happen to satisfy the original Diophantine equation. What we need is the general solution, which now follows. In general, B = -26,277 + 90k    (using the coefft of E) E = 47,022 – 161k    (using the coefft of B) If you substitute these expressions into the original Diophantine equation, the k-terms will cancel out, leaving you with the original eqn. We know that B and E must be smallish numbers and that neither can be negative. For example, if no earrings were sold, Shawn would need to sell over 8 belts to clear $69, and if no belts were sold, then Shawn would need to sell over 15 earrings. So Shawn needs to sell between 0 and 8 belts and between 0 and 15 earrings. Setting k = 292, B = -26,277 + 90*292 = 3 E = 47,022 – 161*292 = 10 B = 3, E = 10 If k is greater than or less than 292 by 1, or more, then either B will be negative or E will be negative. So this is the answer. Check 3*8.05 + 10*4.50 = 25.15 + 45.00 = 69.15 – Correct! Answer: Shawn sells 3 belts and 10 earrings
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What is the diameter of a spiral coil of .65265 inch diameter pipe 100 feet long?

The equation of a spiral in polar coordinates has the general form r=A+Bø, where A is the starting radius of the spiral and B is a factor governing the growth of the spiral outwards. For example, if B=0, there is no outward growth and we just have a circle of radius A. A horizontal line length A represents the initial r, and the angle ø is the angle between r and this horizontal line. So r increases in length as ø increases (this angle is measured in radians where 2(pi) radians = 360 degrees, so 1 radian is 180/(pi)=57.3 degrees approximately.) If B=1/2 and A=5", for example, the minimum radius would be 5" when ø=0. When ø=2(pi) (360 degrees), r=5+(pi), or about 8.14". This angle would bring r back to the horizontal position, but it would be 8.14" instead of the initial 5". At ø=720 degrees, the horizontal line would increase by a further 3.14". Everywhere on the spiral the spiral arms would be 3.14" apart. What would B be if the spiral arms were 0.65625" apart? 2(pi)B=0.65625, so B=0.65625/(2(pi))=0.10445". The equation of the spiral is r=5+0.10445ø. To calculate the length of the spiral we have two possible ways: an approximate value based on the similarity between concentric circles and a spiral; or an accurate value obtainable through calculus. The approximate way is to add together the circumferences of the concentric circles: L=2(pi)(5+(5+0.65625)+...+(5+0.65625N)) where L=spiral length and N is the number of turns. L=2(pi)(5N+0.65625S) where S=0+1+2+3+...+(N-1)=N(N-1)/2. This formula arises from the fact that the first and last terms (0, N-1) the second and penultimate terms (1, N-2) and so on add up to N-1. So, for example, if N were 10 we would have (0+9)+(1+8)+(2+7)+(3+6)+(4+5)=5*9=45=10*9/2. If N were 5 we would have 0+1+2+3+4=10=(0+4)+(1+3)+2=5*4/2. L=12*100 inches. L=1200=2(pi)(5N+0.65625N(N-1)/2)=(pi)N(10+0.65625(N-1))=(pi)N(9.34375+0.65625N). If the external radius is r1 and the internal radius is r then the thickness of the spiral is r1-r and since 0.65625 is the gap between the spiral arms N=(r1-r)/0.65625. N is an integer, but, since it is unlikely that this equation would actually produce an integer we would settle for the nearest integer. If we solve this equation for N, we can deduce the external radius and diameter of the spiral: N(9.34375+0.65625N)=1200/(pi)=381.97; 0.65625N^2+9.34375N-381.97=0 and N=(-9.34375+sqrt(1089.98))/1.3125=18 (nearest integer). This means that there are 18 turns of the spiral to make the total length about 100 feet. If X is the final external diameter of the coiled pipe and the internal radius is 5" (the minimum allowable) then X/2 is the external radius, so N=((X/2)-5)/0.65625. We found N=18 so we can find X: X=2*(0.65625*18+5)=33.625in. Solution using calculus Using calculus, we can work out the relationship between the length of the spiral and other parameters. We start with any polar equation r(ø) and a picture: draw a line representing a general value of r. At a small angle dø to this line we draw another line a little bit longer, length r+dr. Now we join the ends together to make a narrow-angled triangle AOB where angle AOB=dø and AB=ds, the small section of the curve. In the triangle AO is length r and BO is length r+dr. If we mark the point C along BO so that CO is length r, the same as AO, we have an isosceles triangle COA. Because the apex angle is small, CA=rdø, the length of the arc of the sector. In triangle ABC, CB=dr, AB=ds and CA=rdø. By Pythagoras, AB^2=CB^2+CA^2, that is, ds^2=dr^2+r^2dø^2, because angle BCA is a right angle as dø tends to zero. The length of the curve is the result of adding the tiny ds values together between limits of r or ø. We can write ds=sqrt(dr^2+r^2dø^2). If we divide both sides by dr, we get ds/dr=sqrt(1+(rdø/dr)^2) so s=integral(sqrt(1+(rdø/dr)^2)dr, where s is the length of the curve. The integral is definite if we define the limits of r. For our spiral we have r=A+Bø, making ø=(r-A)/B and B=p/(2(pi)), where p is the diameter of the pipe=0.65625", so we can substitute for ø in the integral and the limits for r are A to X/2, where A is the inner radius (A=5") and X/2 is the outer radius. dø/dr=2(pi)/p, a constant=9.57 approx. s=integral(sqrt(1+(2(pi)r/p)^2)dr) between limits r=A to X/2. After the integral is calculated, we solve for X putting s=1200". The expression (2(pi)r/p)^2 is large compared to 1, so s=integral((2(pi)r/p)dr) approximately and s=[(pi)r^2/p] (r=A to X/2); therefore, since we know s=1200, we can write ((pi)/p)(X^2/4-A^2)=1200. Therefore X=2sqrt(1200p/(pi))+A^2)=33.21". Compare this answer with the one we got before and we can see they are close. [We could get a formal solution to the integral, using hyperbolic trigonometric or other logarithmic functions, but such a solution would make it very difficult or tedious to solve for X, since X would appear in logarithmic expressions and in other expressions making it difficult or impossible to isolate X. For example, the next term in the expansion of the integral would be (p/(4(pi))ln(X/2A), having a value of about 0.06. It is anticipated, therefore, that an approximation would be sufficient in this problem with the given figures.] We can feel justified in using the formula for finding the length of pipe, L, when X=6'=72": L=((pi)/p)(1296-25)=6084.52"=507' approximately. This length of pipe would hold 507/100*0.96 gallons=4.87 gallons.      
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x=4 y=12, . Show how you built the equations using your integers.

x + y = 16 x - y = -8 Step 1) Choose two random numbers between -12 and 12 (4 and 12 in this case). Step 2) Let one of the numbers be x and the other be y. Step 3) Add both numbers together to form 1 equation. Step 4) Subtract one number from another to form the other equation. And with that, we are done.
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How can I know how many solutions has the trigonometric equation?

The trig functions sine, cosine, tangent, etc., are cyclic. For example, if sin(x)=1/2, x=30 degrees is only one solution out of an infinite number because x=150, 390, 510, 750, etc., are also solutions. If sin(30x)=1/2, then x=5, 13, 17, 25, etc. Unless there is a specific restraint in the trig equation, there will always be an infinite number of solutions. For example, x=sin(x) has only solution, x=0. Another example is x=2sin(x), which has only three solutions. Sometimes a question will state the limits of the answer. For example, find x between the limits 0 to 360, or 0 to 2(pi), depending on whether x is measured in radians or degrees. In this range there are usually two solutions. Graphically it is easier to tell how many solutions there will be. For example, x=2sin(x) can be represented by two functions plotted together on the same graph: y=x and y=2sin(x). The line y=x intersects the curve y=2sin(x) at 3 points, so there must be 3 solutions to x=2sin(x). You can see this, because the sine curve has humps, like the Loch Ness monster, that peak at 2 and -2 all the way along the x axis in both directions, positive and negative forever. The line joins the origin to (1,1), (2,2), etc. on the positive side and (-1,-1), (-2,-2), etc. on the negative side, and it cuts the sine curve through the first negative hump and the first positive hump including the origin, making three points of intersection only, because the line just rises over the humps after that. I hope this helps you to answer your trig questions.
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how many ways are there to add and get the sum of 180

There are an infinite number of ways to get 180 from two numbers, if we count decimals and fractions as well as other real numbers; but if we are limited to positive integers greater than zero and just the sum of two of them, we are limited to x and 180-x. If we also exclude 90+90 because the numbers are the same, then we have 1 to 89 combined with 179 to 91, which is 89 pairs. Moving on to the sum of three different numbers, let's make 1 plus another two different numbers adding up to 179. So we have 2+177, 3+176, ..., 87+92, 88+91, 89+90, which is 88 groups combined with 1. Move on to 2 plus another two different numbers adding up to 178: 3+175, ..., 87+91, 88+90, which is 86 groups. Then we move on to 3 plus 177: 4+173, ..., 86+91, 87+90, 88+89, 85 groups. And so on, with reducing numbers, until we get to 59, 60 and 61. Let's divide the numbers into two groups A and B. In A we start with 1 and in B we put 2 and (180-A-B)=177 as a pair (2,177). Then we put the next pair in group B: (3,176), then (4,175) and keep going till we have used up all the numbers, ending up with (88,90). Then we count how many pairs there are in group B and pair it up with the number in group A, so we start with (1,88) which covers all the combinations of numbers in group B. Now we move to 2 in group A, put all the pairs adding up to 178 in group B, and finally put the count of these pairs with 2 in group A: (2,86). We then move on to 3, and so on, putting in the counts to make up the number pair in group A. When we've finished by putting the last count in group A, which is (59,1), we can forget about group B and look at the pattern in group A. What we see is this: (1,88), (2,86), (3,85), (4,83), (5,82), (6,80), (7,79), ... See how the counts come in pairs with a gap? All the multiples of 3 are missing in the counts sequence (e.g., 87, 84, 81). We find there are 29 pairs and one odd count, 88, which is unpaired. Number the pairs 0 to 28 and refer to the pair number as N. Add the counts in the pairs together so we start with pair 0 as 86+85=171, pair 1 as 165, pair 2 as 159, and so on. The sequence 171, 165, 159, ..., 3 is an arithmetic sequence with a start of 171 and a difference of 6 between each term in the sequence. [Note also that the terms in the series are all multiples of 3: 3*57, 3*55, 3*55, ...] The rule for the Nth term is 171-6N. When N=0 we have the first term 171 and when N=28 the last term is 3. There is one more term at the end which is unpaired made up of the numbers 59, 60 and 61. We can combine this with the unpaired (1,88). We can find the sum of the terms in the series, which will tell us how many ways there are of adding three different integers so that their sum is 180 (like the sum of the angles of a triangle).  To find the sum of the terms of the series we note that there are 29 terms (0 to 28) and they all contain 171, so that's 171*29=4959. We also have to subtract 6(0+1+2+3+...+28)=6*28*29/2=2436. So 4959-2436=2523. [The sum of the series is also 3(57+55+53+...+5+3+1)=2523.] To this we add the "odd couple" 88+1=89 and 2523+89=2612. Add also the 89 which is the number of pairs of integers adding up to 180 we calculated at the beginning. The total so far is 2612+89=2701 ways of adding 2 or 3 positive integers so that their sum is 180. If you want to go further, please feel free to do so!
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how many gallons of a 10% solution must you add to 30 gallons of a 40% solution to get a final solution that is at 25% level of strength

For the purpose of answering this type of question I consider a chemical solution to consist of two materials in liquid form. Then I think of the materials as each being represented by a quantity of balls of the same colour. One of the balls (say, a quantity of white balls of water) is the solvent and the other, the solute, is represented by a quantity of black balls. The strength of the solution is the percentage of black balls in the mixture. Call the quantity of each w and b. A 10% solution consists of 9v/10 white balls and v/10 black balls, where v is the volume, so we can write 9vw/10+bv/10=v(9w/10+b/10) as the 10% solution of v gallons. We can also write 30(60w/100+40b/100)=30(3w/5+2w/5) as 30 gallons of the 40% solution, so 30(3w/5+2b/5)+v(9w/10+b/10)=(v+30)(3w/4+b/4), which is the 25% strength solution, where there is 75% solvent and 25% solute. This equation simplifies to 18w+12b+v(9w/10+b/10)=(v+30)(3w/4+b/4) Multiply through by 20 to get rid of the fractions: 360w+240b+2v(9w+b)=5(v+30)(3w+b). This expands to 360w+240b+18vw+2vb=15vw+5vb+450w+150b. We're looking for v, so we get all the terms containing v together: 3vw-3vb=90w-90b, so v=90(w-b)/(3(w-b))=30. Therefore, 30 gallons of 10% solution are required to reduce the 30 gallons of 40% solution to 25% strength.    
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how to solve this quadratic equation x^2-2^x-13=0

I think you may have typed this in wrongly because the term 2^x is not a permitted term in a quadratic equation. As it stands the solution to the equation is about -3.6168. Let's say you made a mistake in typing and the middle term is 2*x or 2x, then the solution exists but it is irrational (involves square roots). The easiest way to solve x^2-2x-13=0 is to complete the square: (x^2-2x+1)-1-13=0, which is (x-1)^2-14=0, so (x-1)^2=14. Take square roots of each side: x-1=+sqrt(14)=+3.7417 approx. A square root always has two solutions, one positive and one negative, but the same magnitude number. So x has two possible values: x=1+3.7417=4.7417 or x=1-3.7417=-2.7417. If you meant the question to be x^2-12x-13=0, then the solution is (x-13)(x+1)=0 and x=13 or -1. To work this out we ask: what are the factors of 1 (x^2 coefficient) and 13 (the constant term). The factors of 1 are 1 and 1 because only 1 times 1 make 1; and the factors of 13 are only 1 and 13. The coefficient of the x^2 term tells is how many x's go in each bracket. That's 1x or just x in each bracket. And the factors of 13 tell us what. Numbers to write in each bracket, so that's 1 and 13. So we have (x 1)(x 13). What about the signs between them? We look at the sign in front of 13 in the quadratic. It's minus, and that means there will be a plus in one bracket and a minus in the other. But which way round? Well, there's one more test: we take the factors of 13 and subtract them because the minus sign in front of 13 tells us we need to subtract. If it had been plus, we would have added the factors. 13-1=12. If 12 is the coefficient of the x term then the quadratic can be solved. (If the number had not been 12 we could not have solved the quadratic this way.) The sign in front of 12x is the sign that goes in front of the larger number in the brackets, so minus goes in front of 13. So we have (x+1)(x-13)=0. One or other of these factors is zero, so x+1 or x-13 is zero. x+1=0 means x=-1 and x-13=0 means x=13. These are the solutions or roots.
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What is a stem and leaf plot and when would I use one?

Stem and Leaf plots are just a method of ordering data in a dataset to produce a frequency chart. These plots are used in statistical analysis to draw conclusions about a dataset. The usual way this is done is to use part of each datum to create a data bin. Let's imagine a dataset where all the data consists of numbers between 1 and 99. It doesn't matter how big the dataset is or if there are duplicates. Now imagine 10 bins. The first bin is for numbers between 1 and 9; the second for numbers between 10 and 19, and so on. The numbers of the bins will be labelled 0 to 9. The bins are the stems. So we just go through all the data and put each datum into its appropriate bin. But we don't have to put the whole of the data into each bin, because the bin number is already numbered with the first digit of the data. So the contents of each bin just contain the second digit of the data. The bins (stems) are lined up in order 0 to 9 and we can also stack their contents so that the single digits are in order inside the bins. These are the leaves. Imagine the bins are made of glass. We can look at the bins and the heights of the stacks of contents. The heights of the contents form a shape as we run down the line of bins. These heights tell us how many data there are in each bin and indicate where the most data is and where the least data is. This is is a frequency distribution. It's the basis of the Stem and Leaf plot and can be represented by a table or chart. Each row of the table starts with the bin number (STEM) and along the row we have the contents of the bin (LEAVES). Turn the table on its side and we have a chart with the stem running along the bottom and the leaves forming towers over the stems. The chart resembles the row of bins with the stack, or column, of contents over them, but the bins are now invisible, and only their labels remain as regular horizontal divisions on the chart. But it doesn't stop there. This frequency chart tells us where most of the data can be found, where its middle is and the general shape of the data. These are important statistical observations. Not all the bins may have data in them, and some will have lots of data. Random data will produce no particular shape, but in many cases there will be a pattern. We've considered numbers from 1 to 99, but the data can have any range as long as the data is binned carefully to reflect the relative magnitude of the data. If the data were between 250 and 400, for example, we might take the first 2 digits as the bin label: 25 to 40 and the contents would be the third digit. So you need to make a decision based on the range of data values to decide how the data is going to be binned. I hope this helps you to understand Stem and Leaf plots.
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