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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 problems14 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>\{g5 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> (for8 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(15 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 that5 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 = 26 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 have5 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 add5 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 cas6 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 <math4 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