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- * Techniques for Solving Equations ** [[System of equations]]2 KB (198 words) - 17:47, 3 November 2021
- ...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) - 02:00, 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:15 KB (2,396 words) - 20:24, 21 February 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) - 19:01, 29 October 2021
- ....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) - 18:53, 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>\{g4 KB (792 words) - 00:29, 13 April 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) - 21:53, 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) - 04:52, 8 May 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 (1,585 words) - 03:58, 1 May 2023
- ..., 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) - 23:06, 20 November 2023
- 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) - 21:36, 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 = 1</math> and <math>x^2 + y^2 = 2 + \sq5 KB (874 words) - 22:30, 1 April 2022
- 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) - 23:21, 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) - 15:18, 14 July 2023
- 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) - 17:36, 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 (750 words) - 22:55, 5 February 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) - 02:00, 13 January 2024
- ...math>f(1) = 5</math>, and <math>f(2) = 13</math>, we get a system of three equations in three variables: Plugging in <math>c=1</math> into the last two equations gives7 KB (988 words) - 15:14, 10 April 2024
- ...years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages? Find the value of <math>c</math> such that the system of equations30 KB (4,794 words) - 23:00, 8 May 2024
- ...th> of his bill and Joe tipped <math>20\%</math> of his bill. What was the difference, in dollars between their bills? The equations <math> 2x + 7 = 3 </math> and <math> bx - 10 = -2 </math> have the same sol14 KB (2,026 words) - 11:45, 12 July 2021