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  • * Techniques for Solving Equations ** [[System of equations]]
    2 KB (198 words) - 15:06, 7 December 2024
  • ...y two consecutive terms is constant. This constant is called the '''common difference''' of the sequence. ...d <math>4, 12, 36, 108, \ldots</math> are not arithmetic sequences, as the difference between consecutive terms varies.
    4 KB (736 words) - 01:00, 7 March 2024
  • ...the integers modulo 5. In modulo 5, two integers are congruent when their difference is a [[multiple]] of 5. Adding the two equations we get:
    16 KB (2,406 words) - 07:56, 10 July 2024
  • ...'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...roblems, including finding solutions to [[Diophantine equation|Diophantine equations]], testing whether certain large numbers are prime, and even some problems
    14 KB (2,317 words) - 18:01, 29 October 2021
  • There are <math>2007</math> equations. Sum them. We get: ...th>, which is either <math>3</math> or <math>-3</math>. It does not make a difference which one we choose, so we can choose <math>3</math> for convenience. Now,
    8 KB (1,332 words) - 13:39, 8 December 2024
  • ....35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?
    13 KB (1,987 words) - 17:53, 10 December 2022
  • ...and (3), we have <math>m | d-gr^{n+2}+gr^{n+1}</math>. Reinterpreting both equations, To begin, we let the common difference of <math>\{a_n\}</math> be <math>d</math> and the common ratio of <math>\{g
    5 KB (883 words) - 00:05, 2 June 2024
  • ...onsecutive integers whose sum is <math>m.</math> The absolute value of the difference between the greatest element of <math>A</math> and the greatest element of ...th>A</math> has half the number of elements as set <math>B</math>, and the difference between the greatest terms of the two two sequences is <math>99</math> (for
    8 KB (1,437 words) - 20:53, 19 May 2023
  • Rewrite the system of equations as <cmath>\frac{x^{2}}{t-1}+\frac{y^{2}}{t-3^{2}}+\frac{z^{2}}{t-5^{2}}+\fr ...ls on each side are equal at <math>t=4,16,36,64</math>, we can express the difference of the two polynomials by a quartic polynomial that has roots at <math>t=4,
    6 KB (1,051 words) - 18:35, 1 August 2024
  • ...h> Since the coefficient of <math>x</math> must be zero, this gives us two equations, <math>F_{16}b + F_{17}a = 0</math> and <math>F_{15}b + F_{16}a + 1 = 0</ma ...<math>\frac{ax^3+bx^2+1}{x^2-x-1}</math>, we get the following systems of equations:
    10 KB (1,595 words) - 15:30, 24 August 2024
  • ..., 2b, 3b, 4b</math>. Our method will be to use the given numbers to set up equations to solve for <math>a</math> and <math>b</math>, and then calculate <math>(* ...ue of <math>148 - 3a</math>. On the third column from the left, the common difference is <math>103 - 2b</math>, so that square also has a value of <math>2b + 3(1
    5 KB (878 words) - 17:49, 8 November 2024
  • Notice that these factors can be grouped into a difference of squares: ...cevians' length, Applying Stewart theorem to them, getting three different equations:
    14 KB (2,234 words) - 22:46, 30 June 2024
  • ...difference of squares we get <math>d(2a-d) = 264</math>. Subtracting both equations gives <math>2d^2 = 32</math>, <math>d = 4</math>, and <math>a = 35</math>.
    2 KB (320 words) - 14:50, 12 September 2024
  • The sequence <math>\Delta(\Delta A)</math> is the second finite difference sequence, and the first <math>k-1</math> terms of this sequence can be comp Adding the above <math>k-1</math> equations we find that
    5 KB (778 words) - 20:36, 3 December 2022
  • The [[Trigonometric identities|cosine difference identity]] simplifies that to ...ered at <math>(0,0)</math> with radius <math>\sqrt{2+\sqrt{3}}</math>. The equations of these circles are <math>(x-1)^2 + y^2 = 1</math> and <math>x^2 + y^2 = 2
    6 KB (932 words) - 14:55, 16 November 2024
  • Denote the first term as <math>a</math>, and the common difference between the first three terms as <math>d</math>. The four numbers thus are ...th>400+x^2=y^2</math>, where <math>y</math> is an integer. Factoring using difference of squares, we have
    5 KB (921 words) - 22:21, 22 January 2023
  • ...>f(3)=1848</math>. Plugging in the values for x gives us a system of three equations: ...er to find <math>f(8)</math> add <math>f(3)</math> enough times to get the difference between the <math>d_1d_2</math> and <math>ad_2+bd_1</math> terms, then add
    5 KB (793 words) - 18:55, 10 July 2024
  • Now, if we let <math>z = y + \frac{1}{y}</math>, we can get the equations By the difference of cubes formula, <math>2(1-y^3)=2(1-y)(1+y+y^2)</math>, so we have two cas
    6 KB (1,060 words) - 16:36, 26 April 2024
  • ...implies that <math>a^2 + b^2 = 1^2 + 7^2 = 50</math>. Combining these two equations yields ...y-coordinates of <math>C</math> and <math>D'</math> are, respectively, the difference between the x-coordinates and the y-coordinates of <math>A</math> and <math
    4 KB (748 words) - 18:41, 30 November 2024
  • ...lways read the problem VERY carefully before attempting; it could mean the difference of making the cutoff. ...irst term say <math>a</math>. Since the numbers are consecutive the common difference <math>d = 1</math>.
    3 KB (450 words) - 01:00, 13 January 2024

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